# 11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

Size: px
Start display at page:

Download "11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space"

Transcription

1 11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of a vector.! Perform vector operations and interpret the results geometrically.! Write a vector as a linear combination of standard unit vectors. Component Form of a Vector! Use vectors to solve problems involving force or velocity. 3 4 Component Form of a Vector Component Form of a Vector Many quantities in geometry and physics, such as area, volume, temperature, mass, and time, can be characterized by a single real number scaled to appropriate units of measure. These are called scalar quantities, and the real number associated with each is called a scalar. Other quantities, such as force, velocity, and acceleration, involve both magnitude and direction and cannot be characterized completely by a single real number. 5 A directed line segment is used to represent such a quantity, as shown in Figure The directed line segment has initial point P and terminal point Q, and its length (or magnitude) is denoted by Directed line segments that have the same length and direction are equivalent, as shown in Figure Figure 11.1 Figure

2 Component Form of a Vector The set of all directed line segments that are equivalent to a given directed line segment is a vector in the plane and is denoted by v =. In typeset material, vectors are usually denoted by lowercase, boldface letters such as u, v, and w. When written by hand, however, vectors are often denoted by letters with arrows above them, such as and Example 1 Vector Representation by Directed Line Segments Let v be represented by the directed line segment from (0, 0) to (3, 2), and let u be represented by the directed line segment from (1, 2) to (4, 4). Show that v and u are equivalent. Let P(0, 0) and Q(3, 2) be the initial and terminal points of v, and let R(1, 2) and S(4, 4) be the initial and terminal points of u, as shown in Figure Figure Example 1 Solution Example 1 Solution You can use the Distance Formula to show that have the same length. and and Slope of Slope of Both line segments have the same direction, because they both are directed toward the upper right on lines having the same slope. Because and have the same length and direction, you can conclude that the two vectors are equivalent. That is, v and u are equivalent Component Form of a Vector Component Form of a Vector The directed line segment whose initial point is the origin is often the most convenient representative of a set of equivalent directed line segments such as those shown in Figure This representation of v is said to be in standard position. A directed line segment whose initial point is the origin can be uniquely represented by the coordinates of its terminal point Q(v 1, v 2 ), as shown in Figure Figure Figure

3 Component Form of a Vector Component Form of a Vector The following procedures can be used to convert directed line segments to component form or vice versa. 1. If P(p 1, p 2 ) and Q(q 1, q 2 ) are the initial and terminal points of a directed line segment, the component form of the vector v represented by is = This definition implies that two vectors u = and v = are equal if and only if u1 = v1 and u2 = v2. Moreover, from the Distance Formula you can see that the length (or magnitude) of is v is Component Form of a Vector 2. If v =, v can be represented by the directed line segment, in standard position, from P(0, 0) to Q(v 1, v 2 ). The length of v is called the norm of v. If, v is a unit vector. Moreover, if and only if v is the zero vector 0. Vector Operations Vector Operations Vector Operations Geometrically, the scalar multiple of a vector v and a scalar c is the vector that is c times as long as v, shown in Figure If c is positive, cv has the same direction as v. If c is negative, cv has the opposite direction. 17 Figure

4 Vector Operations Vector Operations The sum of two vectors can be represented geometrically by positioning the vectors (without changing their magnitudes or directions) so that the initial point of one coincides with the terminal point of the other, as shown in Figure The vector u + v, called the resultant vector, is the diagonal of a parallelogram having u and v as its adjacent sides. Figure 11.8 shows the equivalence of the geometric and algebraic definitions of vector addition and scalar multiplication, and presents (at far right) a geometric interpretation of u v. Figure Figure Example 3 Vector Operations Example 3 Solution Given v = and w =, find each of the vectors. c. Using 2w = you have a. b Vector Operations Vector Operations Any set of vectors (with an accompanying set of scalars) that satisfies the eight properties given in Theorem 11.1 is a vector space. The eight properties are the vector space axioms. So, this theorem states that the set of vectors in the plane (with the set of real numbers) forms a vector space

5 Vector Operations Vector Operations Generally, the length of the sum of two vectors is not equal to the sum of their lengths.! To see this, consider the vectors u and v as shown in Figure In Theorem 11.3, u is called a unit vector in the direction of v. The process of multiplying v by to get a unit vector is called normalization of v. 25 Figure Vector Operations By considering u and v as two sides of a triangle, you can see that the length of the third side is u + v, and you have!!! u + v! u + v. Equality occurs only if the vectors u and v have the same direction.! Standard Unit Vectors This result is called the triangle inequality for vectors Standard Unit Vectors The unit vectors and are called the standard unit vectors in the plane and are denoted by as shown in Figure These vectors can be used to represent any vector uniquely, as follows. Standard Unit Vectors The vector v = v 1 i + v 2 j is called a linear combination of i and j. The scalars v 1 and v 2 are called the horizontal and vertical components of v. Figure

6 Example 5 Writing a Linear Combination of Unit Vectors Let u be the vector with initial point (2, 5) and terminal point ( 1, 3), and let v = 2i j. Write each vector as a linear combination of i and j. a. u b. w = 2u 3v a. u = Applications of Vectors b. w = Applications of Vectors Vectors have many applications in physics and engineering. One example is force. A vector can be used to represent force, because force has both magnitude and direction. Example 7 Finding the Resultant Force Two tugboats are pushing an ocean liner, as shown in Figure Each boat is exerting a force of 400 pounds. What is the resultant force on the ocean liner? If two or more forces are acting on an object, then the resultant force on the object is the vector sum of the vector forces. 33 Figure Example 7 Solution Using Figure 11.12, you can represent the forces exerted by the first and second tugboats as F 1 = = F 2 = = Example 7 Solution The resultant force on the ocean liner is F = F 1 + F 2 = = So, the resultant force on the ocean liner is approximately 752 pounds in the direction of the positive x-axis

7 Applications of Vectors In surveying and navigation, a bearing is a direction that measures the acute angle that a path or line of sight makes with a fixed north-south line.! In air navigation, bearings are measured in degrees clockwise from north.! 11.2 Space Coordinates and Vectors in Space 37 Copyright Cengage Learning. All rights reserved. 38 Objectives! Understand the three-dimensional rectangular coordinate system.! Analyze vectors in space. Coordinates in Space! Use three-dimensional vectors to solve real-life problems Coordinates in Space Before extending the concept of a vector to three dimensions, you must be able to identify points in the three-dimensional coordinate system. You can construct this system by passing a z-axis perpendicular to both the x-and y-axes at the origin. Figure shows the positive portion of each coordinate axis. Taken as pairs, the axes determine three coordinate planes: the xy-plane, the xz-plane, and the yz-plane. Coordinates in Space These three coordinate planes separate three-space into eight octants. The first octant is the one for which all three coordinates are positive. In this three dimensional system, a point P in space is determined by an ordered triple (x, y, z) where x, y, and z are as follows. x = directed distance from yz-plane to P y = directed distance from xz-plane to P z = directed distance from xy-plane to P Figure Several points are shown in Figure Figure

8 Coordinates in Space A three-dimensional coordinate system can have either a left-handed or a right-handed orientation. To determine the orientation of a system, imagine that you are standing at the origin, with your arms pointing in the direction of the positive x- and y-axes, and with the z-axis pointing up, as shown in Figure Coordinates in Space Many of the formulas established for the two-dimensional coordinate system can be extended to three dimensions. For example, to find the distance between two points in space, you can use the Pythagorean Theorem twice, as shown in Figure The system is right-handed or left-handed depending on which hand points along the x-axis. In this text, you will work exclusively with the right-handed system. Figure Figure Coordinates in Space Example 1 Finding the Distance Between Two Points in Space By doing this, you will obtain the formula for the distance between the points (x 1, y 1, z 1 ) and (x 2, y 2, z 2 ). The distance between the points (2, 1, 3) and (1, 0, 2) is Coordinates in Space Coordinates in Space A sphere with center at (x 0, y 0, z 0 ) and radius r is defined to be the set of all points (x, y, z) such that the distance between (x, y, z) and (x 0, y 0, z 0 ) is r. If (x, y, z) is an arbitrary point on the sphere, the equation of the sphere is You can use the Distance Formula to find the standard equation of a sphere of radius r, centered at (x 0, y 0, z 0 ). as shown in Figure Moreover, the midpoint of the line segment joining the points (x 1, y 1, z 1 ) and (x 2, y 2, z 2 ). has coordinates Figure

9 Vectors in Space Vectors in Space In space, vectors are denoted by ordered triples v = v 1, v 2, v 3. The zero vector is denoted by 0 = 0, 0, 0. Using the unit vectors i = 1, 0, 0, j = 0, 1, 0, and k = 0, 0,1 in the direction of the positive z-axis, the standard unit vector notation for v is as shown in Figure Figure Vectors in Space Vectors in Space If v is represented by the directed line segment from P (p 1, p 2, p 3 ) to Q(q 1, q 2, q 3 ), as shown in Figure 11.20, the component form of v is given by subtracting the coordinates of the initial point from the coordinates of the terminal point, as follows. v = v 1, v 2, v 3 = q 1 p 1, q 2 p 2, q 3 p 3 Figure Example 3 Finding the Component Form of a Vector in Space Find the component form and magnitude of the vector v having initial point ( 2, 3,1) and terminal point (0, 4, 4). Then find a unit vector in the direction of v. Example 3 Solution The unit vector in the direction of v is The component form of v is v = q 1 p 1, q 2 p 2, q 3 p 3 = 0 ( 2), 4 3, 4 1 = 2, 7, 3 which implies that its magnitude is 53 54

10 Vectors in Space The definition of scalar multiplication that positive scalar multiples of a nonzero vector v have the same direction as v, whereas negative multiples have the direction opposite of v. In general, two nonzero vectors u and v are parallel if there is some scalar c such that u = cv. Vectors in Space For example, in Figure 11.21, the vectors u, v and w are parallel because u = 2v and w = v. Figure Example 4 Parallel Vectors Example 4 Solution Vector w has initial point (2, 1, 3) and terminal point ( 4, 7, 5). Which of the following vectors is parallel to w? a. u = 3, 4, 1 b. v = 12, 16, 4 Begin by writing w in component form. w = 4 2, 7 ( 1), 5 3 = 6, 8, 2 a. Because u = 3, 4, 1 = 6, 8, 2 = w, you can conclude that u is parallel to w. b. In this case, you want to find a scalar c such that 12, 16, 4 = c 6, 8, = 6c c = 2 16 = 8c c = 2 4 = 2c c = 2 Because there is no c for which the equation has a solution, the vectors are not parallel The Dot Product of Two Vectors Objectives! Use properties of the dot product of two vectors.! Find the angle between two vectors using the dot product.! Find the direction cosines of a vector in space.! Find the projection of a vector onto another vector.! Use vectors to find the work done by a constant force. Copyright Cengage Learning. All rights reserved. 60

11 The Dot Product The Dot Product You have studied two operations with vectors vector addition and multiplication by a scalar each of which yields another vector. In this section you will study a third vector operation, called the dot product. This product yields a scalar, rather than a vector The Dot Product Example 1 Finding Dot Products Given, and, find each of the following Example 1 Solution Angle Between Two Vectors Notice that the result of part (b) is a vector quantity, whereas the results of the other three parts are scalar quantities

12 Angle Between Two Vectors Angle Between Two Vectors The angle between two nonzero vectors is the angle!, 0!!! ", between their respective standard position vectors, as shown in Figure The next theorem shows how to find this angle using the dot product. If the angle between two vectors is known, rewriting Theorem 11.5 in the form Figure produces an alternative way to calculate the dot product Angle Between Two Vectors From this form, you can see that because " u " and " v " are always positive, u. v and cos! will always have the same sign. Figure shows the possible orientations of two vectors. Angle Between Two Vectors From Theorem 11.5, you can see that two nonzero vectors meet at a right angle if and only if their dot product is zero. Two such vectors are said to be orthogonal. Figure Example 2 Finding the Angle Between Two vectors For u = 3, 1, 2, v = 4, 0,2, w = 1, 1, 2, and z = 2, 0, 1, find the angle between each pair of vectors. a. u and v b. u and w c. v and z Example 2 Solution Because u. w = 0, u and w are orthogonal. So,! = "/2. Because u. v < 0, Consequently,! = ". Note that v and z are parallel, with v = 2z

13 Direction Cosines In space it is convenient to measure direction in terms of the angles between the nonzero vector v and the three unit vectors i, j, and k, and as shown in Figure Direction Cosines 73 Figure Direction Cosines The angles #, \$ and % are the direction angles of v, and cos #, cos \$, and cos % are the direction cosines of v. Direction Cosines By similar reasoning with the unit vectors j and k, you have Because and it follows that cos # = v 1 / v Direction Cosines Consequently, any non zero vector v in space has the normalized form and because v/ v is a unit vector, it follows that Example 3 Finding Direction Angles Find the direction cosines and angles for the vector v = 2i + 3j + 4k, and show that cos 2 # + cos 2 \$ + cos 2 % = 1. Because you can write the following

14 Example 3 Solution Furthermore, the sum of the squares of the direction cosines is Projections and Vector Components See Figure Figure Projections and Vector Components You have already seen applications in which two vectors are added to produce a resultant vector. Many applications in physics and engineering pose the reverse problem decomposing a given vector into the sum of two vector components. Projections and Vector Components Consider a boat on an inclined ramp, as shown in Figure The force F due to gravity pulls the boat down the ramp and against the ramp. These two forces, w 1 and w 2, are orthogonal they are called the vector components of F. The following physical example enables you to see the usefulness of this procedure. 81 Figure Projections and Vector Components Projections and Vector Components The forces w 1 and w 2 help you analyze the effect of gravity on the boat. For example, w 1 indicates the force necessary to keep the boat from rolling down the ramp, whereas w 2 indicates the force that the tires must withstand. 83 Figure

15 Example 4 Finding a Vector Component of u Orthogonal to v Example 4 Solution Find the vector component of to, given that and that is orthogonal Check to see that w 2 is orthogonal to v, as shown in Figure Because u = w 1 + w 2, where w 1 is parallel to v, it follows that w 2 is the vector component of u orthogonal to v. So, you have 85 Figure Projections and Vector Components The projection of u onto v can be written as a scalar multiple of a unit vector in the direction of v.that is, 11.4 The Cross Product of Two Vectors in Space The scalar k is called the component of u in the direction of v. 87 Copyright Cengage Learning. All rights reserved. 88 Objectives! Find the cross product of two vectors in space.! Use the triple scalar product of three vectors in space. The Cross Product 89 90

16 The Cross Product The Cross Product Many applications in physics, engineering, and geometry involve finding a vector in space that is orthogonal to two given vectors. You will study a product that will yield such a vector. It is called the cross product, and it is most conveniently defined and calculated using the standard unit vector form. Because the cross product yields a vector, it is also called the vector product The Cross Product A convenient way to calculate u # v is to use the following determinant form with cofactor expansion. The Cross Product Note the minus sign in front of the j-component. Each of the three 2 # 2 determinants can be evaluated by using the following diagonal pattern. Here are a couple of examples Example 1 Finding the Cross Product Given u = i 2j + k and v = 3i + j 2k, find each of the following. a. u # v b. v # u c. v # v Example 1 Solution Note that this result is the negative of that in part (a)

17 The Cross Product The Cross Product The Cross Product Both u # v and v # u are perpendicular to the plane determined by u and v. One way to remember the orientations of the vectors u, v and u # v is to compare them with the unit vectors i, j, and k = i # j, as shown in Figure Example 2 Using the Cross Product Find a unit vector that is orthogonal to both u = i 4j + k and v = 2i + 3j. The cross product u # v, as shown in Figure 11.37, is orthogonal to both u and v. Figure The three vectors u, v and u # v form a right-handed system, whereas the three vectors u, v, and v # u form a left-handed system. 99 Figure Example 2 Solution Because a unit vector orthogonal to both u and v is The Cross Product In physics, the cross product can be used to measure torque the moment M of a force F about a point P, as shown in Figure If the point of application of the force is Q, the moment of F about P is given by The magnitude of the moment M measures the tendency of the vector to rotate counterclockwise (using the right-hand rule) about an axis directed along the vector M. 101 Figure

18 The Triple Scalar Product The Triple Scalar Product For vectors u, v, and w in space, the dot product of u and v # w u! (v # w) is called the triple scalar product, as defined in Theorem The Triple Scalar Product The Triple Scalar Product If the vectors u, v, and w do not lie in the same plane, the triple scalar product u! (v # w) can be used to determine the volume of the parallelepiped (a polyhedron, all of whose faces are parallelograms) with u, v, and w as adjacent edges, as shown in Figure Figure Example 5 Volume by the Triple Scalar Product Find the volume of the parallelepiped shown in Figure having u = 3i 5j + k, v = 2j 2k, and w = 3i + j + k as adjacent edges. Example 5 Solution By Theorem 11.10, you have Figure

19 The Triple Scalar Product A natural consequence of Theorem is that the volume of the parallelepiped is 0 if and only if the three vectors are coplanar. That is, if the vectors 11.5 Lines and Planes in Space have the same initial point, they lie in the same plane if and only if 109 Copyright Cengage Learning. All rights reserved. 110 Objectives! Write a set of parametric equations for a line in space.! Write a linear equation to represent a plane in space.! Sketch the plane given by a linear equation. Lines in Space! Find the distances between points, planes, and lines in space Lines in Space In Figure 11.43, consider the line L through the point (x 1, y 1, z 1 ) and parallel to the vector. The vector v is a direction vector for the line L, and a, b, and c are direction numbers. One way of describing the line L is to say that it consists of all points Q(x, y, z) for which the vector is parallel to v. P Lines in Space This means that is a scalar multiple of v, and you can write = t v, where t is a scalar (a real number). By equating corresponding components, you can obtain parametric equations of a line in space. Figure

20 Lines in Space Example 1 Finding Parametric and Symmetric Equations Find parametric and symmetric equations of the line L that passes through the point (1, 2, 4 ) and is parallel to If the direction numbers a, b, and c are all nonzero, you can eliminate the parameter t to obtain symmetric equations of the line. To find a set of parametric equations of the line, use the coordinates x 1 = 1, y 1 = 2, and z 1 = 4 and direction numbers a = 2, b = 4, and c = 4 (see Figure 11.44). 115 Figure Example 1 Solution x = 1 + 2t, y = 2 + 4t, z = 4 4t Because a, b, and c are all nonzero, a set of symmetric equations is Planes in Space Planes in Space Planes in Space Consider the plane containing the point P(x 1, y 1, z 1 ) having a nonzero normal vector n =, as shown in Figure This plane consists of all points Q(x, y, z) for which vector is orthogonal to n. Using the dot product, you can write the following. By regrouping terms, you obtain the general form of the equation of a plane in space. The third equation of the plane is said! to be in standard form. Figure Given the general form of the equation of a plane, it is easy to find a normal vector to the plane. Simply use the coefficients of x, y, and z and write n =. 120

21 Example 3 Finding an Equation of a Plane in Three-Space Example 3 Solution Find the general equation of the plane containing the points (2, 1, 1), (0, 4, 1), and ( 2, 1, 4). To apply Theorem you need a point in the plane and a vector that is normal to the plane. There are three choices for the point, but no normal vector is given. To obtain a normal vector, use the cross product of vectors u and v extending from the point (2, 1, 1) to the points (0, 4, 1) and ( 2, 1, 4), as shown in Figure The component forms of u and v are and it follows that! Figure Example 3 Solution Planes in Space is normal to the given plane. Using the direction numbers for n and the point (x 1, y 1, z 1 ) = (2, 1, 1), you can determine an equation of the plane to be!!a(x x 1 ) + b(y y 1 ) + c(z z 1 ) = 0!! 9(x 2) + 6(y 1) + 12(z 1) = 0!!! 9x + 6y + 12z 36 = 0!!! 3x + 2y + 4z 12 = 0.! Two distinct planes in three-space either are parallel or intersect in a line. If they intersect, you can determine the angle between them from the angle (0!!! "/2) between their normal vectors, as shown in Figure Figure Planes in Space Specifically, if vectors n 1 and n 2 are normal to two intersecting planes, the angle! between the normal vectors is equal to the angle between the two planes and is given by Sketching Planes in Space Consequently, two planes with normal vectors n 1 and n 2 are 1. perpendicular if n 1! n 2 = parallel if n 1 is a scalar multiple of n

22 Sketching Planes in Space If a plane in space intersects one of the coordinate planes, the line of intersection is called the trace of the given plane in the coordinate plane. To sketch a plane in space, it is helpful to find its points of intersection with the coordinate axes and its traces in the coordinate planes. Sketching Planes in Space For example, consider the plane given by 3x + 2y + 4z = 12. You can find the xy-trace by letting z = 0 and sketching the line in the xy-plane. 3x + 2y = 12 This line intersects the x-axis at (4, 0, 0) and the y-axis at (0, 6, 0) Sketching Planes in Space Sketching Planes in Space In Figure 11.49, this process is continued by finding the yztrace and the xz-trace, and then shading the triangular region lying in the first octant.! If an equation of a plane has a missing variable, such as 2x + z = 1, the plane must be parallel to the axis represented by the missing variable, as shown in Figure ! Figure Figure Sketching Planes in Space If two variables are missing from an equation of a plane, it is parallel to the coordinate plane represented by the missing variables, as shown in Figure ! Distances Between Points, Planes, and Lines Figure

23 Distances Between Points, Planes, and Lines This section is concluded with the following discussion of two basic types of problems involving distance in space. 1. Finding the distance between a point and a plane 2. Finding the distance between a point and a line Distances Between Points, Planes, and Lines If P is any point in the plane, you can find this distance by projecting the vector onto the normal vector n. The length of this projection is the desired distance.! The distance D between a point Q and a plane is the length of the shortest line segment connecting Q to the plane, as shown in Figure Figure Example 5 Finding the Distance Between a Point and a Plane Find the distance between the point Q(1, 5, 4 ) and the plane given by 3x y + 2z = 6. Example 5 Solutions Using the Distance Formula given in Theorem produces You know that is normal to the given plane. To find a point in the plane, let y = 0 and z = 0 and obtain the point P(2, 0, 0). The vector from P to Q is given by Distances Between Points, Planes, and Lines Distances Between Points, Planes, and Lines From Theorem 11.13, you can determine that the distance between the point Q(x 0, y 0, z 0 ) and the plane given by ax + by + cz + d = 0 is or where P(x 1, y 1, z 1 ) is a point in the plane and d = (ax 1 + by 1 + cz 1 )

24 11.6 Surfaces in Space Objectives! Recognize and write equations of cylindrical surfaces.! Recognize and write equations of quadric surfaces.! Recognize and write equations of surfaces of revolution. Copyright Cengage Learning. All rights reserved Cylindrical Surfaces You have already studied two special types of surfaces. Cylindrical Surfaces A third type of surface in space is called a cylindrical surface, or simply a cylinder Cylindrical Surfaces To define a cylinder, consider the familiar right circular cylinder shown in Figure Cylindrical Surfaces This circle is called a generating curve for the cylinder, as indicated in the following definition. You can imagine that this cylinder is generated by a vertical line moving around the circle x 2 + y 2 = a 2 in the xy-plane. Figure Figure

25 Cylindrical Surfaces Cylindrical Surfaces For the right circular cylinder shown in Figure 11.56, the equation of the generating curve is x 2 + y 2 = a 2. Equation of generating curve in xy- plane To find an equation of the cylinder, note that you can generate any one of the rulings by fixing the values of x and y and then allowing z to take on all real values. In this sense, the value of z is arbitrary and is, therefore, not included in the equation. In other words, the equation of this cylinder is simply the equation of its generating curve. x 2 + y 2 = a 2 Equation of cylinder in space Example 1 Sketching a Cylinder Example 1 Solution Sketch the surface represented by each equation. a. z = y 2 b. z = sin x, 0! x & 2" a. The graph is a cylinder whose generating curve, z = y 2, is a parabola in the yz-plane. b. The graph is a cylinder generated by the sine curve in the xz-plane. The rulings are parallel to the y-axis, as shown in Figure 11.58(b). The rulings of the cylinder are parallel to the x-axis, as shown in Figure11.58(a). Figure 11.58(a) 147 Figure 11.58(b) 148 Quadric Surfaces The fourth basic type of surface in space is a quadric surface. Quadric surfaces are the three-dimensional analogs of conic sections. Quadric Surfaces

26 Quadric Surfaces Quadric Surfaces The intersection of a surface with a plane is called the trace of the surface in the plane. To visualize a surface in space, it is helpful to determine its traces in some wellchosen planes. The traces of quadric surfaces are conics. These traces, together with the standard form of the equation of each quadric surface, are shown in the following table Quadric Surfaces Quadric Surfaces Example 2 Sketching a Quadric Surface Classify and sketch the surface given by 4x 2 3y z = 0. Begin by writing the equation in standard form. Example 2 Solution To sketch the graph of this surface, it helps to find the traces in the coordinate planes. You can conclude that the surface is a hyperboloid of two sheets with the y-axis as its axis

27 Example 2 Solution The graph is shown in Figure Surfaces of Revolution Figure Surfaces of Revolution The fifth special type of surface you will study is called a surface of revolution. You will now look at a procedure for finding its equation. Consider the graph of the radius function y = r(z) Generating curve in the yz-plane. Surfaces of Revolution If this graph is revolved about the z-axis, it forms a surface of revolution, as shown in Figure The trace of the surface in the plane z = z 0 is a circle whose radius is r(z 0 ) and whose equation is x 2 + y 2 = [r(z 0 )] 2. Circular trace in plane: z = z 0 Replacing z 0 with z produces an equation that is valid for all values of z. 159 Figure Surfaces of Revolution In a similar manner, you can obtain equations for surfaces of revolution for the other two axes, and the results are summarized as follows. Example 5 Finding an Equation for a Surface of Revolution a. An equation for the surface of revolution formed by revolving the graph of about the z-axis is

28 Example 5 Finding an Equation for a Surface of Revolution b. To find an equation for the surface formed by revolving the graph of 9x 2 = y 3 about the y-axis, solve for x in terms of y to obtain So, the equation for this surface is 11.7 Cylindrical and Spherical Coordinates The graph is shown in Figure Figure Copyright Cengage Learning. All rights reserved. 164 Objectives! Use cylindrical coordinates to represent surfaces in space.! Use spherical coordinates to represent surfaces in space. Cylindrical Coordinates Cylindrical Coordinates The cylindrical coordinate system, is an extension of polar coordinates in the plane to three-dimensional space. Cylindrical Coordinates To convert from rectangular to cylindrical coordinates (or vice versa), use the following conversion guidelines for polar coordinates, as illustrated in Figure Figure

29 Cylindrical Coordinates Example 1 Converting from Cylindrical to Rectangular Coordinates Cylindrical to rectangular: Rectangular to cylindrical: Convert the point (r,!, z) = coordinates. Using the cylindrical-to-rectangular conversion equations produces to rectangular The point (0, 0, 0) is called the pole. Moreover, because the representation of a point in the polar coordinate system is not unique, it follows that the representation in the cylindrical coordinate system is also not unique. 169 So, in rectangular coordinates, the point is (x, y, z) = as shown in Figure Figure Cylindrical Coordinates Cylindrical coordinates are especially convenient for representing cylindrical surfaces and surfaces of revolution with the z-axis as the axis of symmetry, as shown in Figure Cylindrical Coordinates Vertical planes containing the z-axis and horizontal planes also have simple cylindrical coordinate equations, as shown in Figure Figure Figure Spherical Coordinates In the spherical coordinate system, each point is represented by an ordered triple: the first coordinate is a distance, and the second and third coordinates are angles. Spherical Coordinates This system is similar to the latitude-longitude system used to identify points on the surface of Earth

30 Spherical Coordinates Spherical Coordinates For example, the point on the surface of Earth whose latitude is 40 North (of the equator) and whose longitude is 80 West (of the prime meridian) is shown in Figure Assuming that the Earth is spherical and has a radius of 4000 miles, you would label this point as Figure Spherical Coordinates The relationship between rectangular and spherical coordinates is illustrated in Figure To convert from one system to the other, use the following. Spherical Coordinates To change coordinates between the cylindrical and spherical systems, use the following. Spherical to cylindrical (r! 0): Spherical to rectangular: Cylindrical to spherical (r! 0): Rectangular to spherical: Figure Spherical Coordinates The spherical coordinate system is useful primarily for surfaces in space that have a point or center of symmetry. For example, Figure shows three surfaces with simple spherical equations. Example 5 Rectangular-to-Spherical Conversion Find an equation in spherical coordinates for the surface represented by each rectangular equation. a. Cone: x 2 + y 2 + z 2 b. Sphere: x 2 + y 2 + z 2 4z = 0 a. Making the appropriate replacements for x, y, and z in the given equation yields the following. Figure

31 Example 5 Solution Example 5 Solution b. Because and the given equation has the following spherical form. The equation ' = "/4 represents the upper half-cone, and the equation ' = 3"/4 represents the lower half-cone. Temporarily discarding the possibility that ( = 0, you have the spherical equation Example 5 Solution Note that the solution set for this equation includes a point for which ( = 0, so nothing is lost by discarding the factor (. The sphere represented by the equation ( = 4cos ' is shown in Figure Figure

### Two vectors are equal if they have the same length and direction. They do not

Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

### Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### 13.4 THE CROSS PRODUCT

710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### Figure 1.1 Vector A and Vector F

CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### 6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

### Module 8 Lesson 4: Applications of Vectors

Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

### Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

### MATH 275: Calculus III. Lecture Notes by Angel V. Kumchev

MATH 275: Calculus III Lecture Notes by Angel V. Kumchev Contents Preface.............................................. iii Lecture 1. Three-Dimensional Coordinate Systems..................... 1 Lecture

### Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

### The Dot and Cross Products

The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and

### A vector is a directed line segment used to represent a vector quantity.

Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector

### Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50

Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall

### Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi

Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in three-dimensional space, we also examine the

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

### 1.5 Equations of Lines and Planes in 3-D

40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from

### ... ... . (2,4,5).. ...

12 Three Dimensions ½¾º½ Ì ÓÓÖ Ò Ø ËÝ Ø Ñ So far wehave been investigatingfunctions ofthe form y = f(x), withone independent and one dependent variable Such functions can be represented in two dimensions,

### Vector Algebra CHAPTER 13. Ü13.1. Basic Concepts

CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have

### 9 Multiplication of Vectors: The Scalar or Dot Product

Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

### Unit 11 Additional Topics in Trigonometry - Classwork

Unit 11 Additional Topics in Trigonometry - Classwork In geometry and physics, concepts such as temperature, mass, time, length, area, and volume can be quantified with a single real number. These are

### Geometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v

12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The

### Surface Normals and Tangent Planes

Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to

### discuss how to describe points, lines and planes in 3 space.

Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position

### 42 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2. 1.6.1 Brief review of Conic Sections

2 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more sense. You

### MAT 1341: REVIEW II SANGHOON BAEK

MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and

### THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

### Vector Algebra II: Scalar and Vector Products

Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define

### Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

### 12.5 Equations of Lines and Planes

Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### Concepts in Calculus III

Concepts in Calculus III Beta Version UNIVERSITY PRESS OF FLORIDA Florida A&M University, Tallahassee Florida Atlantic University, Boca Raton Florida Gulf Coast University, Ft. Myers Florida International

### One advantage of this algebraic approach is that we can write down

. Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the x-axis points out

### CHAPTER FIVE. 5. Equations of Lines in R 3

118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Problem set on Cross Product

1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j - 3 k ) 2 Calculate the vector product of i - j and i + j (Ans ) 3 Find the unit vectors that are perpendicular

### Dot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product

Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,

### PHYSICS 151 Notes for Online Lecture #6

PHYSICS 151 Notes for Online Lecture #6 Vectors - A vector is basically an arrow. The length of the arrow represents the magnitude (value) and the arrow points in the direction. Many different quantities

### Exam 1 Sample Question SOLUTIONS. y = 2x

Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

### Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

### Review Sheet for Test 1

Review Sheet for Test 1 Math 261-00 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### An Application of Analytic Geometry to Designing Machine Parts--and Dresses

Electronic Proceedings of Undergraduate Mathematics Day, Vol. 3 (008), No. 5 An Application of Analytic Geometry to Designing Machine Parts--and Dresses Karl Hess Sinclair Community College Dayton, OH

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

### 2.1 Three Dimensional Curves and Surfaces

. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline

### AP Physics - Vector Algrebra Tutorial

AP Physics - Vector Algrebra Tutorial Thomas Jefferson High School for Science and Technology AP Physics Team Summer 2013 1 CONTENTS CONTENTS Contents 1 Scalars and Vectors 3 2 Rectangular and Polar Form

### 28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

### 8-3 Dot Products and Vector Projections

8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### Chapter 11 Equilibrium

11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Section 9.5: Equations of Lines and Planes

Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### Review of Vector Analysis in Cartesian Coordinates

R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.

### Mathematics Notes for Class 12 chapter 10. Vector Algebra

1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative

### KEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007

KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 4-47) Surveys and

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### Vectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.

Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type

### Vector has a magnitude and a direction. Scalar has a magnitude

Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Volumes of Revolution

Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize -dimensional solids and a specific procedure to sketch a solid of revolution. Students

### 1. Vectors and Matrices

E. 8.02 Exercises. Vectors and Matrices A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by dir A = A, (A 0); A it is the unit vector lying along A and pointed like

### Physics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus

Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### Unified Lecture # 4 Vectors

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### VELOCITY, ACCELERATION, FORCE

VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how

### Notes on the representational possibilities of projective quadrics in four dimensions

bacso 2006/6/22 18:13 page 167 #1 4/1 (2006), 167 177 tmcs@inf.unideb.hu http://tmcs.math.klte.hu Notes on the representational possibilities of projective quadrics in four dimensions Sándor Bácsó and

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Essential Mathematics for Computer Graphics fast

John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made

### Section 10.4 Vectors

Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

### Illinois State Standards Alignments Grades Three through Eleven

Illinois State Standards Alignments Grades Three through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other

### Math 215 HW #6 Solutions

Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs