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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "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"

Transcription

1 Homework Solutions 5/ 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, 3. A vector parallel to L 2 is 2, 3, 1. If these vectors are parallel, then the lines are parallel too. So check the ratios of the components: The ratio of the first components is 6 = 3. 2 The ratio of the second components is 9 = 3. 3 The ratio of the third components is 3 = 3. 1 So 6, 9, 3 = 3 2, 3, 1, and these vectors are parallel. Therefore L 1 is parallel to L 2. We could obviously stop here, but I will test for intersection and skewness just for demonstration purposes. If the two lines intersect, there must be a point (x, y, z) that satisfies the equations for L 1 for some value of t, and the equations for L 2 for some value of s. For this value of s and t, we must have 6t = x = 1 + s 1 + 9t = y = 4 3s 3t = z = s From the first equation, we see that t = 1 1 s. The second equation becomes s = 4 3s 3 2 s = 9 2 s = 3 1

2 and thus t = 1 1 = 2. Plugging this into the left side of the third equation gives t = 2 s So the third equation cannot be satisfied. Therefore there is no point where L 1 and L 2 intersect, which is expected because they re parallel. If the test for the lines being parallel had failed, then we would conclude the lines are skew Find the equation of the plane through the origin and parallel to 2x y + 3z = 1. Let P 1 be the plane we re finding the equation for, and P 0 be the plane with equation 2x y + 3z = 1. Then we know P 1 is parallel to P 0 and (0, 0, 0) P 1. To find the equation for P 1, we need a vector normal to P 1 and a point in P 1. We already have the point, so we just need the normal vector. Since P 1 is parallel to P 0, every line L 1 in P 1 must be parallel to some line L 0 in P 0. That means any vector parallel to L 1 is also parallel to L 0. But by definition, any vector normal to P 0 is orthogonal to every vector parallel to L 0. So a vector normal to P 0 must also be normal to P 1. Thus 2, 1, 3 is normal to P 1. So an equation for P 1 is 2(x 0) (y 0) + 3(z 0) = 0 or more compactly 2x y + 3z = Find an equation of the plane containing ( 1, 2, 1) and the line of intersection of the planes x + y z = 2 and 2x y + 3z = 1. We are given a point on the plane, so we need only to find a normal vector. If the point ( 1, 2, 1) is on the line of intersection of the other two planes, then we don t have enough information to get the equation of the plane. Otherwise, we can find a normal vector by taking the cross product of a vector parallel to that line of intersection, and a vector going from ( 1, 2, 1) to some point on that line. (Both of those vectors are parallel to lines in the 2

3 plane we want the equation of, so a vector orthogonal to both of them must be normal to the plane). First, we need to find the line of intersection of the planes x + y z = 2 and 2x y + 3z = 1. This is done by solving for two of the variables in terms of the other one. A point (x, y, z) in the first plane must have z = x + y 2. If this point is also in the second plane, then 2x y + 3x + 3y 6 = 1 5x + 2y = 7 y = x So if we let x = t, then y = 7 5t and z = x + y 2 = 3 3 t. A vector parallel to this line is then 1, 5, The vector 2, 5, 3 is also parallel to this line (it s a scalar multiple of the other one) and is going to be easier to deal with. A point on the line of intersection is ( 0, 7, 3 2 2) which is reached at t = 0. The vector from ( 1, 2, 1) to this point is 1, 3 2, 1 2. This means the vector 2, 3, 1 is also in the plane as it is a scalar multiple. So we take the cross product to get a normal vector: 2, 3, 1 2, 5, 3 = 9 + 5, 2 + 6, 10 6 = 4, 8, 16 Thus, an equation of the plane is 4(x + 1) + 8(y 2) 16(z 1) = 0, or x + 2y 4z = 1 if we distribute and divide through by Find the point at which the line x = 3 t y = 2 + t z = 5t intersects the plane x y + 2z = 9. Just substitute for each variable in the equation of the plane with the expressions from the equation of the line. 3

4 (3 t) (2 + t) + 2(5t) = t = 9 t = 1 So the point is (2, 3, 5) Determine whether the planes P 1 : x = 4y 2z P 2 : 8y = 1 + 2x + 4z are parallel, perpendicular, or neither. If neither, find the angle between them. As in a problem above, we see that planes are parallel if and only if their normal vectors are parallel. An equation in standard form for P 1 is x 4y + 2z = 0, and one for P 2 is 2x + 4z 8y = 1. So a normal vector for P 1 is n := 1, 4, 2 and a normal vector for P 2 is m := 2, 4, 8. Check the ratios of the components: The first component ratio is 1, while the second is 1. So they are not parallel. 2 It can be shown using elementary geometry that the angle between planes is equal to the angle between their normal vectors. So if θ is the angle between P 1 and P 2, then ( ) n m θ = cos 1 n m ( ) = cos ( ) 34 = cos ( ) = cos ( ) 34 = cos 1 42 ( = cos 1 17 )

5 Find parametric equations for the line L 1 through the point (0, 1, 2) that is parallel to the plane P 1 with equation x + y + z = 2 and is perpendicular to the line L 2 with parametric equations x = 1 + t y = 1 t z = 2t For the equation of a line we need a point on the line and a vector parallel to it. So all we need is a vector a parallel to L 1. Since L 1 is parallel to P 1, it must be parallel to some line L 3 in P 1. Any vector parallel to L 1 must be parallel to L 3, and so must be orthogonal to any normal vector of P 1, one of which is 1, 1, 1. So a is orthogonal to 1, 1, 1. Since L 1 is perpendicular to L 2, any vector parallel to L 1 must be orthogonal to any vector parallel to L 2. The vector 1, 1, 2 is parallel to L 2, so we must have a orthogonal to 1, 1, 2 also. Thus a is some multiple of i j k 1, 1, 1 1, 1, 2 = = 2 + 1, 1 + 1, 1 1 = 3, 2, So an equation for the line L 1 is x = 3t y = 1 + 2t z = 2 2t Describe and sketch the surface z = cos(x). This is a cosine cylinder, consisting of lines parallel to the y-axis and having an intersection with the xz-plane that is the cosine curve. 5

6 Find the traces of the surface y 2 = x 2 + z 2 in the planes x = k, y = k, z = k and sketch it. With x = k, we have y 2 z 2 = k 2. This is a hyperbola opening up along the line parallel to the y-axis for k 0. For k = 0 it is the two lines x = 0, y = t, z = t and x = 0, y = t, z = t. With y = k, we have x 2 + z 2 = k 2. This is a circle of radius k for k 0, just the origin for k = 0. With z = k, we have y 2 x 2 = k 2. This is a hyperbola opening up along the line parallel to the y-axis for k 0. For k = 0 it is the two lines x = t, y = t, z = 0 and x = t, y = t, z = 0. 6

7 Since the traces are hyperbolas or pairs of lines in both x = k and z = k planes, and ellipses (or circles) in the y = k planes, this surface is a cone opening along the y-axis Find traces for the surface y = z 2 x 2 and sketch it. With x = k, we have y = z 2 k 2, which is a parabola opening up in the positive-y direction along the line parallel to the y-axis. 7

8 With y = k, we have z 2 x 2 = k, which is a hyperbola opening up along the z-axis for k > 0. For k = 0 it is the pair of lines x = t, y = 0, z = t and x = t, y = 0, z = t. For k < 0 it is a hyperbola opening up along the x-axis. With z = k, we have y = x 2 +k 2, which is a parabola opening up in the negative-y direction along the line parallel to the y-axis. Since the traces are hyperbolas in y = k and parabolas in x = k and z = k, it is a hyperbolic paraboloid opening up along the y-axis, and along the x-axis to the right of the xz-plane, along the z-axis to the left of the xz-plane

9 Reduce the equation x 2 y 2 + z 2 4x 2y 2z + 4 = 0 to one of the standard forms, classify the surface, and sketch it. Completing the square thrice gives ( x 2 4x + 4 ) 4 ( y 2 + 2y + 1 ) ( z 2 2z + 1 ) = 0 (x 2) 2 (y + 1) 2 + (z 1) 2 = 0 (x 2) 2 + (z 1) 2 = (y + 1) 2 This is a cone with center (2, 1, 1), that is, where the tips of the two cones meet, and which opens up around the line x = 2, y = t, z = 1 which is parallel to the y-axis Find an equation for the surface consisting of all points equidistant from ( 1, 0, 0) and the plane x = 1. Classify this surface. In 10.5 we have the distance from a point (x 1, y 1, z 1 ) to the plane ax + by + cz + d = 0 given by 9

10 D = ax 1 + by 1 + cz 1 + d a2 + b 2 + c 2 So if the distance from ( 1, 0, 0) and the plane x = 1 are equal, then x 1 (x + 1) 2 + y 2 + z 2 = = x Squaring both sides gives (x 1) 2 + y 2 + z 2 = (x 1) 2 (x + 1) 2 + y 2 + z 2 = (x 1) 2 y 2 + z 2 = (x 1) 2 (x + 1) 2 y 2 + z 2 = 4x 1 4 y2 1 4 z2 = x which is the equation of an elliptic paraboloid opening in the negative x-direction around the x-axis, with vertex at (0, 0, 0). The point ( 1, 0, 0) is said to be the focus of the paraboloid, and the plane x = 1 is the directrix. 10

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

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

More information

Homework 3 Model Solution Section

Homework 3 Model Solution Section Homework 3 Model Solution Section 12.6 13.1. 12.6.3 Describe and sketch the surface + z 2 = 1. If we cut the surface by a plane y = k which is parallel to xz-plane, the intersection is + z 2 = 1 on a plane,

More information

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

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

More information

1.5 Equations of Lines and Planes in 3-D

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

More information

(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,

(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0, Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We

More information

Midterm Exam I, Calculus III, Sample A

Midterm Exam I, Calculus III, Sample A Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of

More information

Sphere centered at the origin.

Sphere centered at the origin. A Quadratic surfaces In this appendix we will study several families of so-called quadratic surfaces, namely surfaces z = f(x, y) which are defined by equations of the type A + By 2 + Cz 2 + Dxy + Exz

More information

Name: Class: Date: Conics Multiple Choice Post-Test. Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: Conics Multiple Choice Post-Test. Multiple Choice Identify the choice that best completes the statement or answers the question. Name: Class: Date: Conics Multiple Choice Post-Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1 Graph the equation x = 4(y 2) 2 + 1. Then describe the

More information

Solutions to old Exam 1 problems

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

More information

Test Bank Exercises in. 7. Find the intercepts, the vertical asymptote, and the slant asymptote of the graph of

Test Bank Exercises in. 7. Find the intercepts, the vertical asymptote, and the slant asymptote of the graph of Test Bank Exercises in CHAPTER 5 Exercise Set 5.1 1. Find the intercepts, the vertical asymptote, and the horizontal asymptote of the graph of 2x 1 x 1. 2. Find the intercepts, the vertical asymptote,

More information

12.5 Equations of Lines and Planes

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

More information

Algebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2

Algebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2 1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3-dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect

More information

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

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

More information

2 Topics in 3D Geometry

2 Topics in 3D Geometry 2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and

More information

Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t

Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are

More information

Section 9.5: Equations of Lines and Planes

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

More information

Section 11.4: Equations of Lines and Planes

Section 11.4: Equations of Lines and Planes Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R

More information

10.5. Click here for answers. Click here for solutions. EQUATIONS OF LINES AND PLANES. 3x 4y 6z 9 4, 2, 5. x y z. z 2. x 2. y 1.

10.5. Click here for answers. Click here for solutions. EQUATIONS OF LINES AND PLANES. 3x 4y 6z 9 4, 2, 5. x y z. z 2. x 2. y 1. SECTION EQUATIONS OF LINES AND PLANES 1 EQUATIONS OF LINES AND PLANES A Click here for answers. S Click here for solutions. 1 Find a vector equation and parametric equations for the line passing through

More information

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

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

More information

Math 241, Exam 1 Information.

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)

More information

Lecture 14: Section 3.3

Lecture 14: Section 3.3 Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

More information

Chapter 10: Analytic Geometry

Chapter 10: Analytic Geometry 10.1 Parabolas Chapter 10: Analytic Geometry We ve looked at parabolas before when talking about the graphs of quadratic functions. In this section, parabolas are discussed from a geometrical viewpoint.

More information

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. 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

More information

Section 10.2 The Parabola

Section 10.2 The Parabola 243 Section 10.2 The Parabola Recall that the graph of f(x) = x 2 is a parabola with vertex of (0, 0) and axis of symmetry of x = 0: x = 0 f(x) = x 2 vertex: (0, 0) In this section, we want to expand our

More information

Senior Math Circles February 4, 2009 Conics I

Senior Math Circles February 4, 2009 Conics I 1 University of Waterloo Faculty of Mathematics Conics Senior Math Circles February 4, 2009 Conics I Centre for Education in Mathematics and Computing The most mathematically interesting objects usually

More information

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

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

More information

Review Sheet for Test 1

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

More information

Equations Involving Lines and Planes Standard equations for lines in space

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

More information

Extra Problems for Midterm 2

Extra Problems for Midterm 2 Extra Problems for Midterm Sudesh Kalyanswamy Exercise (Surfaces). Find the equation of, and classify, the surface S consisting of all points equidistant from (0,, 0) and (,, ). Solution. Let P (x, y,

More information

Conic Sections in Cartesian and Polar Coordinates

Conic Sections in Cartesian and Polar Coordinates Conic Sections in Cartesian and Polar Coordinates The conic sections are a family of curves in the plane which have the property in common that they represent all of the possible intersections of a plane

More information

2.1 Three Dimensional Curves and Surfaces

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

More information

LINES AND PLANES CHRIS JOHNSON

LINES AND PLANES CHRIS JOHNSON LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance

More information

Pure Math 30: Explained!

Pure Math 30: Explained! www.puremath30.com 45 Conic Sections: There are 4 main conic sections: circle, ellipse, parabola, and hyperbola. It is possible to create each of these shapes by passing a plane through a three dimensional

More information

September 14, Conics. Parabolas (2).notebook

September 14, Conics. Parabolas (2).notebook 9/9/16 Aim: What is parabola? Do now: 1. How do we define the distance from a point to the line? Conic Sections are created by intersecting a set of double cones with a plane. A 2. The distance from the

More information

decide, when given the eccentricity of a conic, whether the conic is an ellipse, a parabola or a hyperbola;

decide, when given the eccentricity of a conic, whether the conic is an ellipse, a parabola or a hyperbola; Conic sections In this unit we study the conic sections. These are the curves obtained when a cone is cut by a plane. We find the equations of one of these curves, the parabola, by using an alternative

More information

Exam 1 Sample Question SOLUTIONS. y = 2x

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

More information

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

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.

More information

Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

More information

Math 21a Review Session for Exam 2 Solutions to Selected Problems

Math 21a Review Session for Exam 2 Solutions to Selected Problems Math 1a Review Session for Exam Solutions to Selected Problems John Hall April 5, 9 Note: Problems which do not have solutions were done in the review session. 1. Suppose that the temperature distribution

More information

Pre Calculus Math 40S: Explained!

Pre Calculus Math 40S: Explained! www.math0s.com 97 Conics Lesson Part I The Double Napped Cone Conic Sections: There are main conic sections: circle, ellipse, parabola, and hyperbola. It is possible to create each of these shapes by passing

More information

Essential Question: What is the relationship among the focus, directrix, and vertex of a parabola?

Essential Question: What is the relationship among the focus, directrix, and vertex of a parabola? Name Period Date: Topic: 9-3 Parabolas Essential Question: What is the relationship among the focus, directrix, and vertex of a parabola? Standard: G-GPE.2 Objective: Derive the equation of a parabola

More information

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

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:

More information

3.6 Cylinders and Quadric Surfaces

3.6 Cylinders and Quadric Surfaces 3.6 Cylinders and Quadric Surfaces Objectives I know the definition of a cylinder. I can name the 6 quadric surfaces, write their equation, and sketch their graph. Let s take stock in the types of equations

More information

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

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

More information

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

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,

More information

Level: High School: Geometry. Domain: Expressing Geometric Properties with Equations G-GPE

Level: High School: Geometry. Domain: Expressing Geometric Properties with Equations G-GPE 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Translate between the geometric

More information

9 Multiplication of Vectors: The Scalar or Dot Product

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

More information

Tangent and normal lines to conics

Tangent and normal lines to conics 4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

More information

Section 2.4: Equations of Lines and Planes

Section 2.4: Equations of Lines and Planes Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y

More information

Unit 9: Conic Sections Name Per. Test Part 1

Unit 9: Conic Sections Name Per. Test Part 1 Unit 9: Conic Sections Name Per 1/6 HOLIDAY 1/7 General Vocab Intro to Conics Circles 1/8-9 More Circles Ellipses 1/10 Hyperbolas (*)Pre AP Only 1/13 Parabolas HW: Part 4 HW: Part 1 1/14 Identifying conics

More information

Equations of Lines and Planes

Equations of Lines and Planes Calculus 3 Lia Vas Equations of Lines and Planes Planes. A plane is uniquely determined by a point in it and a vector perpendicular to it. An equation of the plane passing the point (x 0, y 0, z 0 ) perpendicular

More information

Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

More information

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000) Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving

More information

Hyperboloid of Two Sheets

Hyperboloid of Two Sheets Math 59 Winter 009 Recitation Handout 5: Sketching D Surfaces If you are simply given an equation of the form z = f(x, y) and asked to draw the graph, producing the graph can be a very complicated and

More information

Unit 10: Quadratic Relations

Unit 10: Quadratic Relations Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This

More information

Engineering Geometry

Engineering Geometry Engineering Geometry Objectives Describe the importance of engineering geometry in design process. Describe coordinate geometry and coordinate systems and apply them to CAD. Review the right-hand rule.

More information

Engineering Drawing. Anup Ghosh

Engineering Drawing. Anup Ghosh Engineering Drawing Anup Ghosh Divide a line in n equal segments. Divide a line in n equal segments. Divide a line in n equal segments. Divide a line in n equal segments. Divide a line in n equal segments.

More information

The Parabola and the Circle

The Parabola and the Circle The Parabola and the Circle The following are several terms and definitions to aid in the understanding of parabolas. 1.) Parabola - A parabola is the set of all points (h, k) that are equidistant from

More information

Chapter 10: Topics in Analytic Geometry

Chapter 10: Topics in Analytic Geometry Chapter 10: Topics in Analytic Geometry 10.1 Parabolas V In blue we see the parabola. It may be defined as the locus of points in the plane that a equidistant from a fixed point (F, the focus) and a fixed

More information

The Parabola. By: OpenStaxCollege

The Parabola. By: OpenStaxCollege The Parabola By: OpenStaxCollege The Olympic torch concludes its journey around the world when it is used to light the Olympic cauldron during the opening ceremony. (credit: Ken Hackman, U.S. Air Force)

More information

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 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

More information

Quadratic curves, quadric surfaces

Quadratic curves, quadric surfaces Chapter 3 Quadratic curves, quadric surfaces In this chapter we begin our study of curved surfaces. We focus on the quadric surfaces. To do this, we also need to look at quadratic curves, such as ellipses.

More information

Activities for Algebra II and Pre Calculus

Activities for Algebra II and Pre Calculus Activities for Algebra II and Pre Calculus Laura Harlow Pearland ISD LauraHarlow8@gmail.com Activities Student engagement Increase student understanding Useful and meaningful Exploration of Conic Sections

More information

MA261-A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM

MA261-A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM MA6-A Calculus III 6 Fall Homework Solutions Due 9//6 :AM 9. # Find the parametric euation and smmetric euation for the line of intersection of the planes + + z = and + z =. To write down a line euation,

More information

Solution. a) The line in question has parameterization γ(t) = (0, t, t). Plugging this into the equation of the surface yields

Solution. a) The line in question has parameterization γ(t) = (0, t, t). Plugging this into the equation of the surface yields Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Midterm # 1 (Tue 21 Feb 2012) Practice Exam Solution Guide Practice problems: The following assortment of problems

More information

10-5 Parabolas. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

10-5 Parabolas. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2 10-5 Parabolas Warm Up Lesson Presentation Lesson Quiz 2 Warm Up 1. Given, solve for p when c = Find each distance. 2. from (0, 2) to (12, 7) 13 3. from the line y = 6 to (12, 7) 13 Objectives Write the

More information

MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

More information

SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

More information

( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those

( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those 1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make

More information

Practice Problems for Midterm 2

Practice Problems for Midterm 2 Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,

More information

Applications of Integration Day 1

Applications of Integration Day 1 Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

More information

Solution: 2. Sketch the graph of 2 given the vectors and shown below.

Solution: 2. Sketch the graph of 2 given the vectors and shown below. 7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit

More information

CHAPTER FIVE. 5. Equations of Lines in R 3

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

More information

Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes

Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes Jim Lambers MAT 169 Fall Semester 009-10 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that

More information

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

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

More information

0.1 Linear Transformations

0.1 Linear Transformations .1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called

More information

Use Geometry Expressions to find equations of curves. Use Geometry Expressions to translate and dilate figures.

Use Geometry Expressions to find equations of curves. Use Geometry Expressions to translate and dilate figures. Learning Objectives Loci and Conics Lesson 2: The Circle Level: Precalculus Time required: 90 minutes Students are now acquainted with the idea of locus, and how Geometry Expressions can be used to explore

More information

Math 211, Multivariable Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions

Math 211, Multivariable Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions Math 211, Multivariable Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions For each question, I give a model lution ( you can see the level of detail that I expect from you in the exam) me comments.

More information

Coimisiún na Scrúduithe Stáit State Examinations Commission LEAVING CERTIFICATE 2009 MARKING SCHEME HIGHER LEVEL

Coimisiún na Scrúduithe Stáit State Examinations Commission LEAVING CERTIFICATE 2009 MARKING SCHEME HIGHER LEVEL Coimisiún na Scrúduithe Stáit State Examinations Commission LEAVING CERTIFICATE 2009 MARKING SCHEME DESIGN & COMMUNICATION GRAPHICS HIGHER LEVEL LEAVING CERTIFICATE 2009 DESIGN & COMM. GRAPHICS HIGHER

More information

Surface Normals and Tangent Planes

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

More information

New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

More information

Practice Problems for Midterm 1

Practice Problems for Midterm 1 Practice Problems for Midterm 1 Here are some problems for you to try. A few I made up, others I found from a variety of sources, including Stewart s Multivariable Calculus book. (1) A boy throws a football

More information

Mathematics 1. Lecture 5. Pattarawit Polpinit

Mathematics 1. Lecture 5. Pattarawit Polpinit Mathematics 1 Lecture 5 Pattarawit Polpinit Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance

More information

Geometry Enduring Understandings Students will understand 1. that all circles are similar.

Geometry Enduring Understandings Students will understand 1. that all circles are similar. High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

More information

Advanced Math Study Guide

Advanced Math Study Guide Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular

More information

We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model

We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant

More information

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. 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

More information

MAT 1341: REVIEW II SANGHOON BAEK

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

More information

EER#21- Graph parabolas and circles whose equations are given in general form by completing the square.

EER#21- Graph parabolas and circles whose equations are given in general form by completing the square. EER#1- Graph parabolas and circles whose equations are given in general form by completing the square. Circles A circle is a set of points that are equidistant from a fixed point. The distance is called

More information

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

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

More information

299ReviewProblemSolutions.nb 1. Review Problems. Final Exam: Wednesday, 12/16/2009 1:30PM. Mathematica 6.0 Initializations

299ReviewProblemSolutions.nb 1. Review Problems. Final Exam: Wednesday, 12/16/2009 1:30PM. Mathematica 6.0 Initializations 99ReviewProblemSolutions.nb Review Problems Final Exam: Wednesday, /6/009 :30PM Mathematica 6.0 Initializations R.) Put x@td = t - and y@td = t -. Sketch on the axes below the curve traced out by 8x@tD,

More information

Section 13.5 Equations of Lines and Planes

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.

More information

17.1. Conic Sections. Introduction. Prerequisites. Learning Outcomes. Learning Style

17.1. Conic Sections. Introduction. Prerequisites. Learning Outcomes. Learning Style Conic Sections 17.1 Introduction The conic sections (or conics) - the ellipse, the parabola and the hperbola - pla an important role both in mathematics and in the application of mathematics to engineering.

More information

Solutions for Review Problems

Solutions for Review Problems olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

More information

13.4 THE CROSS PRODUCT

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

More information

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved. 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

More information

-axis -axis. -axis. at point.

-axis -axis. -axis. at point. Chapter 5 Tangent Lines Sometimes, a concept can make a lot of sense to us visually, but when we try to do some explicit calculations we are quickly humbled We are going to illustrate this sort of thing

More information

Notes on the representational possibilities of projective quadrics in four dimensions

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

More information

= y y 0. = z z 0. (a) Find a parametric vector equation for L. (b) Find parametric (scalar) equations for L.

= y y 0. = z z 0. (a) Find a parametric vector equation for L. (b) Find parametric (scalar) equations for L. Math 21a Lines and lanes Spring, 2009 Lines in Space How can we express the equation(s) of a line through a point (x 0 ; y 0 ; z 0 ) and parallel to the vector u ha; b; ci? Many ways: as parametric (scalar)

More information

JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson

JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3

More information