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

Size: px
Start display at page:

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

Transcription

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

2 Launch Mathematica. Type <<Mathetic`vecpack` Instructions for Getting Started hold down the shift key, and press the return key. Wait for Mathematica s response. (Note: be sure to use the ` symbol rather than the '. You may need to hunt for it on your keyboard: on most, it s in the top left corner.) This essential first step sets up Mathematica for this module. If you omit this bit, the special commands (see below) will not work. Mathematica Commands The following Mathematica commands may be useful to you in this module. Commands that come with Mathematica: Dot, Clear, ArcCos, ViewPoint (3D graphics option), Simplify Special commands for this module: MDDot, GiveQuestion, LastAnswer, AngleBetween, VectorSum3D, Magnitude, Cross, CrossPlot For information on, say, the VectorSum3D command, type?vectorsum3d hold down shift, and press return. And, if you wanted a list of all the commands containing the word Vector you could type?*vector*

3 Experiment 1: The scalar product Preparatory reading In Vectors 1 (Experiment 3) we looked at what happens when we multiply a vector by a scalar. Now we want to ask: what does it mean to multiply a vector by a vector? What this means is by no means obvious. We can motivate things with an important physical law. Newton s Principle of Work can be stated as: The work done by a force F in moving through a displacement r at an angle of θ degrees to its line of action is F r cosθ. This will be our definition for the scalar product. Given two vectors a and b with an angle θ between them, their scalar product is: a b = a b cosθ The product is denoted by a dot, we say a dot b, and so it is often also called the dot product. It s called a product because it behaves in some ways like the product operation in arithmetic. And it s called the scalar product rather than just the product of vectors because there s another kind of product, the vector product, which will appear with three-dimensional vectors in a later experiment. We ve defined the special command MDDot to calculate dot products. The vectors are input to the command in magnitude direction form using the curly bracket notation that we used in the Vectors 1 module: MDDot[{1,0}, {2,45}] The answer in this case is a simple form involving a square root. In other cases, you ll need to get things into decimals; you can apply the N command thus: MDDot[{2.3,47}, {5,119}] //N The command:??mddot reveals how it actually works. What do you think the Abs command does? 1) Try out the MDDot command on some vectors of your choice. In particular, try it out on some equal vectors, parallel vectors, negative vectors and perpendicular vectors (e.g. i and j). 2) For a general vector a, use the definition to find a a. What are the following: i i, j j, i j, j i? Note down these results for use in the next experiment. 3) If a b = 0 but neither a nor b is equal to 0 (the zero vector) what must be the angle between a and b? 4) Try to answer the following questions about the properties of the scalar product: (i) is the scalar product commutative: that is, is it true that a b = b a for all vectors a and b?

4 (ii) is the scalar product distributive: that is, is it true that a (b + c) = a b + b c for all vectors a, b and c? (iii) [careful!] is the scalar product associative: that is, is it true that (a b) c = a (b c) for all vectors a, b and c? This section uses this module s special functions. If they fail to work try going back to the Instructions for Getting Started at the beginning, Post-experiment reading If a b = 0 and neither a nor b is zero then the angle between a and b must be 90 degrees. The properties of the scalar product: (i) the scalar product is a number, (ii) a a = a 2, (iii) a b = b a (commutativity), (iv) a (b + c) = a b + b c (distributivity). What about associativity? That is meaningless for the scalar product, since a.b is a scalar and so (a b) c cannot be performed. Practice Questions This module includes a feature which allows you to get Mathematica to generate practice questions and their answers. There is a set of questions for practising the calculation of dot products. To generate a question, type: GiveQuestion["dot product"] not forgetting to shift-return. To generate the answer for checking, type: LastAnswer["dot product"] You can do this as often as you want: the questions are randomly generated, and repetitions should be rare. Note: You don t need to retype these commands for another question; simply click the mouse on the previously-typed command and shift-return. This section uses this module s special functions. If they fail to work try going back to the Instructions for Getting Started at the beginning. Experiment 2: The scalar product by components, and angles between vectors Preparatory reading It is often important to find the angles between two vectors. In two dimensions, it s not hard to convert the vectors into magnitude direction form and then subtract the angles, but in three dimensions, which we move to in the next experiment, there is no single angle for direction and the

5 problem is harder. In both cases we can do the whole thing easily in component form by making use of the scalar product. Mathematica has a built-in command for doing scalar products of vectors in component form, for example: {1, 3}.{4, -1} Using the dot (full stop) character like this is a shorthand for the full command: Dot[{1, 3}, {4, -1}] If you like to work with the i, j Cartesian notation you can define the two unit vectors like this: i={1, 0}; j={0, 1}; Then you can write vectors as i + 3j, 4i - j and so forth. 1) Considering that the scalar product is defined in terms of magnitudes and angles, you may not expect it to work very elegantly in component form. Let s take a look at that now. Try out various scalar products using the Dot command and look for a pattern. In particular try things like: {3, 4}.{1, 0} {0, 1}.{5, 1} You can obtain the general formula by doing the following (the Clear command removes any values previously assigned to the symbols): Clear[a, b, c, d] {a, b}.{c, d} 2) It s not difficult to prove the formula. Remembering that the scalar product is distributive, expand out the brackets in (a i + b j) (c i + d j). Now simplify this using the results you obtained above for i i, i j etc. This proof is given in the Post-experiment Reading. 3) Can you rearrange the scalar product formula from Experiment 1 into a formula for the angle between two vectors? Try out your new formula on vectors where you know what the angle is (something simple like 0 or 90 ). You can use Mathematica to help: the relevant commands are Dot, Magnitude and ArcCos. You may also need to use N to get answers into decimal form. We ve defined a special command called AngleBetween to do this calculation. For example: AngleBetween[{1,3}, {3,-1}] 4) Considering the angle between formula, can you see a quick way for spotting that two vectors are perpendicular (that is, the angle between them is 90 degrees)? Post-experiment reading Given any two vectors whose components are a = a 1 i + a 2 j, b = b 1 i+ b 2 j, then

6 a b = (a 1 i + a 2 j) (b 1 i + b 2 j) = a 1 b 1 i i + a 1 b 2 i j+ a 2 b 1 j i + a 2 b 2 j j = a 1 b 1 (1)+ a 1 b 2 (0) +a 2 b 1 (0)+ a 2 b 2 (1) = a 1 b 1 + a 2 b 2. In words, then, the rule for doing scalar products in component form is: multiply together the corresponding components, and add up the answers. The formula for the angle between two vectors is a simple rearrangement of the scalar product formula: a b = abcosθ cosθ = a b ab θ = arccos a b ab It is often very important to know whether two (non-zero) vectors are perpendicular or not (whether the angle between them is 90 degrees). To do this, we only need to check if a b = 0 we need not go to the trouble of finding out the vector magnitudes as well. Practice Questions There is a set of questions on the calculation of angles between vectors (expressed in component form). To get a question type: GiveQuestion["angle between"] and to generate the answer type: LastAnswer["angle between"] You can do this as often as you want: the questions are randomly generated, and repetitions should be rare. Note: You don t need to retype these commands for another question; simply click the mouse on the command and shift-return. Experiment 3: Vectors in three dimensions i, j and k Preparatory reading You should be familiar with the fact that any point in three-dimensional space can be represented by three coordinates, (x, y, z). In a similar way, vectors in three dimensional space can be represented by three perpendicular components in the directions of the x, y and z axes. We can write down a 3-D vector in column form, for example: 2 3 4

7 The Cartesian basis (see Vectors 1, Experiment 6) can be extended from two into three dimensions by introducng a third basis vector, k, which is one unit long in the z direction, in addition to i and j. This diagram shows the vector 2i + 3j + 4k: We won t here be using any magnitude direction form for 3-D vectors, although it is possible to do so: the complication is that two angles are required to specify the direction of a vector, which makes the trigonometry pretty hard. Adding vectors in three dimensions works in just the same way as in two: simply add the components. For example: (2i + 3j + 4k) + (3i 5j + k) = 5i 2j + 5k. The same geometrical constructs for vector addition, the Triangle Rule and Parallelogram Rule, can be applied in three dimensions as in two. Since we will always be working in component form for 3- D vectors we won t (fortunately) need to use the Rules for calculating vector additions, but it s still important to be aware geometrically of what s happening in the addition process. The command VectorSum3D is the 3-D equivalent of the command VectorSum. It only works with vectors in component form, for reasons discussed in the Preparatory Reading. Example: VectorSum3D[{{2,0,0}, {0,3,0}, {0,0,4}}, ResultQ->True] It is convenient to define the i, j and k unit vectors in Mathematica: i={1,0,0}; j={0,1,0}; k={0,0,1}; Then you can use these to write vectors in the form 2i + 3j + 4k, for example. We will assume that these definitions are in effect from now on. You can look at how the Triangle and Parallelogram Rules work in 3-D by specifying the following options:

8 VectorSum3D[{2i + 3j + 4k, 3i - 5j + k}, ResultQ->True] VectorSum3D[{2i + 3j + 4k, 3i - 5j + k}, AddRule->Parallelogram] 1) The following command draws a cube (with labels and box removed for clarity): VectorSum3D[{i,j,-i,-j,k,i,j,-i,-j,i,-k,j,k,-i,-k}, LabelQ->False, Boxed->False] Use VectorSum3D to draw: a cube whose diagonals are connected; a pyramid; a tetrahedron. Try drawing some other shapes that take your fancy. 2) We often need to view 3-D pictures from different viewpoints. To do this, Mathematica provides an option called ViewPoint, which can be used with all the 3-D plotting commands we ll be using in this module. Here s an example: VectorSum3D[{i, j, -i-j}, ViewPoint->{0,0,2}] Make a sketch of the output. Try to work out what ViewPoint is doing by substituting different values into the command above, making a sketch for each. Try these other viewpoints : {0,0,-2}, {2,0,0}, {2,0,0.5}, {-2,0,0.5}, {0,2,0.5}. Try some more viewpoints, and other 3D plots, of your own. Here s how it works: ViewPoint defines the point in space from which the 3-D object is to be viewed, using a coordinate system in which {0, 0, 0} is the centre of the viewing box (hence these are not the actual coordinates of the image). Some useful values: {-2,0,0} from the side, looking in the positive x direction {4,0,0} in the negative x direction, and further away {0,-2,0} in the positive y direction {0,0,2} from above (i.e. in the negative z direction) {1.3,-2.4,2} the default (standard) viewpoint 3) Use VectorSum3D to investigate: (i) can you find two non-zero vectors in 3-D which add up to give 0? (ii) Can you find three non-zero vectors in 3-D which add up to give 0? 4) Given any two vectors a and b, investigate the set of vectors made by adding scalar multiples of a and b (3a + 7b for example). Do these vectors all lie on the same surface in space? If so, what is that surface? Begin your investigation with the special cases in which a and b are two of the Cartesian basis vectors i, j or k. 5) Can you work out the magnitude of the vector 2i + 3j + 4k? (Hint: you need to construct two right-angled triangles and use Pythagoras theorem in each.) The answer is given in the Postexperiment Reading. Check your answer with: Magnitude[2i + 3j + 4k] Can you devise a general formula for the magnitude of any vector x i + y j + z k? You can get the answer by doing: Clear[x, y, z]; Magnitude[x i + y j + z k]

9 Post-experiment reading This diagram shows the geometrical construction required to determine v, the magnitude of the vector 2i + 3j + 4k: In the lower right-angled triangle, with sides of length 2, 3 and w, Pythagoras theorem tells us: w 2 = and then in the upper right-angled triangle, with sides of length w, 4 and v, again by Pythagoras theorem: Hence the magnitude is v = 29. v 2 = w = = 29. The magnitude of any vector having components xi + yj + zk is x 2 + y 2 + z 2. (You could verify this simply by replacing the numbers 2, 3 and 4 in the above diagram with x, y and z.) Experiment 4: The scalar product and angles between vectors in three dimensions Preparatory reading In this experiment we look at the scalar product in three dimensions. The definition for the scalar product of two three-dimensional vectors a and b is, as it was in two dimensions: a b = a b cosθ where θ is the angle between the vectors. One way to think about this is that a and b lie in a (unique) plane in 3-D space, and within that plane we are once again in a 2-D space. The formula for the angle

10 between two three-dimensional vectors is, as before, a simple rearrangement of the scalar product formula: θ = cos 1 where a and b denote the magnitudes of a and b. This formula gives us a quick test for perpendicularity of vectors: if the dot product of two (non-zero) vectors is zero, those vectors are perpendicular, otherwise they cannot be. a b ab Mathematica s Dot command will calculate the scalar product of 3-D vectors in the same way as it did for 2-D vectors: {1,3,2}.{4,-1,0} You should define i, j and k in Mathematica (see the last experiment) so that the above product may be written: (i + 3j + 2k).(4i - j) 1) Using the Dot command, try to identify the general formula for dot products in 3-D, bearing in mind the 2-D formula. Consider inputs like: i.(j + k) (3i).(4i + j + k) Try out some inputs of your own along these lines. The following command will reveal the general formula (you will need to Clear the variables x1, etc. if you have used any of them already): (x1 i + y1 j+ z1 k).(x2 i+ y2 j+ z2 k) You may like to prove the formula by expanding out the brackets in the expression ( x 1 i + y 1 j+ z 1 k) ( x 2 i + y 2 j+z 2 k) and simplifying according to the values of i i, i j, i k etc. 2) Given any two (non-zero) vectors a and b, draw a diagram to illustrate how the angles between the vectors a, b, a and b are related. Confirm your findings using the AngleBetween command, which may be used to determine the angle between two 3-D vectors for example: AngleBetween[i, i+k] AngleBetween[-i, i+k] 3) The following command constructs the figure of a tetrahedron by specifying its edges as vectors: VectorSum3D[{i+Sqrt[3]*j, i-sqrt[3]*j, -2i, i+j/sqrt[3]+2k*sqrt[2/3], 2j/Sqrt[3]-2k*Sqrt[2/3], i-sqrt[3]*j, -i+j/sqrt[3]+2k*sqrt[2/3]}, Boxed->False, LabelQ->False]

11 Confirm that this is indeed a tetrahedron by checking, using Magnitude, that the lengths of all the edges are equal, and, using AngleBetween, that the angles between intersecting edges are all 60. (Using the copy-and-paste feature of the Mathematica notebook will save you a lot of repeated typing here.) Post-experiment reading The general rule for calculating the scalar product of 3-D vectors in component form is the same as it was for 2-D vectors: multiply together the corresponding components, and add up the answers. The following diagram shows how the angles between the vectors a, a, b and b are related: One can read off that, for example, (angle between a and b) = 180 (angle between a and b). Practice Questions There are three sets of questions which revise topics from two dimensions for the case of threedimensional vectors. To generate a question from the first set, type GiveQuestion["addition 3D"] And to generate the answer for checking, type LastAnswer["addition 3D"] The other sets of questions are accessed using GiveQuestion["dot product 3D"] GiveQuestion["angle between 3D"]

12 Experiment 5: The vector product Preparatory reading In this experiment we return again to the question of the possible meaning of multiplying a vector with a vector. This time we ll define a product whose outcome is a vector rather than a scalar, hence it is called the vector product. As with the scalar product we ll motivate the definition with a situation in mechanics where the product appears naturally. You may recall from your work in mechanics this definition for the moment of a force in two dimensions: Given a force F which acts through a point P, whose position vector is r, the moment of F about the origin O is defined to be where θ is the angle between r and F. r F sin θ P θ F r O In two dimensions we need to state whether the moment is a clockwise or anticlockwise one. In the above diagram a clockwise moment is shown, one which would cause a clockwise rotation if the system were pivoted at O. In three dimensions the situation is more complicated because we need to state not only the sense (clockwise or anticlockwise) of the moment, but also the direction of the axis about which it acts: that is, the axis about which the system would rotate if it were able to do so freely. Hence a moment in three dimensions possesses both a magnitude, r F sin θ, and a direction: the direction of the axis of rotation. In short, it must be a vector quantity. In the above diagram, if a three-dimensional moment is represented then we can say that it shows a moment T of magnitude r F sin θ which is directed at right angles to the paper. Our argument so far is deficient in one vital respect: consider what the problem is, and how it might be solved, before reading any further. The problem is: there are two directions at right angles to the paper: into it, and out of it. We have to specify which. How can we do that?

13 That s a bit of a trick question, because there is no natural solution to the problem. We simply have to make a choice an arbitrary one and stick to it from here on. The universally-accepted rule is this: choose the direction of T so that the triple of vectors (r, F, T) forms a right-handed system. This is a difficult idea; here s what it means: (i) Imagine yourself with a screwdriver in your right hand. (ii) Approach the vectors r and F in such a way that when you turn the screwdriver in the tightening (clockwise) direction it turns from r to F. (iii) Then the direction in which the screwdriver is pointing is the direction of T. So, then, according to this rule what is the direction of the moment T in the diagram above? Answer: into the paper, as the following diagram demonstrates: P r O from r to F F The vector moment is one example of a general product of vectors. In general, given two vectors a and b, at an angle θ from one another, we define their vector product as: a b = ( a b sinθ)ˆ n where n ˆ denotes the unit vector which is perpendicular to both a and b in such a way that the triple of vectors (a, b, n ˆ ) forms a right-handed system. Vector product is often also called the cross product, because of the cross symbol used in its notation. Another notation you may sometimes see, which means exactly the same thing, is a b. We offer two special commands for doing vector, or cross, products in Mathematica. The first simply calculates the vector product, with i j written as: Cross[i, j] The second command both calculates the answer and draws a 3-D graph: CrossPlot[i, j] The input vectors are shown in red and blue, the product vector in purple, and the plane in which the input vectors lie marked in grey, to emphasise that the product vector is perpendicular to both of the inputs.

14 1) Calculate, by hand, all the possible vector products involving the Cartesian basis vectors i, j and k: i i, i j, j i, i k, etc. Remember that the angle between any two of the basis vectors is 90 degrees, which simplifies things a lot. You ll also need to know that the vector triple (i, j, k) forms a right-handed system. Be warned: there is a difference between i j and j i! Check your answers using Cross or CrossPlot. 2) Use Cross or CrossPlot to try to answer these question about the vector product: (i) What is the vector product of any vector a with itself, a a? (ii) Is the vector product commutative: that is, is it true that a b = b a for all vectors a and b? Your work in part 1 should answer both of these straight away. The next two are harder though (iv especially): (iii) Is the vector product distributive: that is, is it true that a (b + c) = a b + b c for all vectors a, b and c? (iv) Is the vector product associative: that is, is it true that (a b) c = a (b c) for all vectors a, b and c? [Hint: if you want to answer these questions in general using Mathematica, it is useful to define three general vectors, such as: a = a1 i + a2 j + a3 k; b = b1 i + b2 j + b3 k; c = c1 i + c2 j + c3 k; Then you can easily build-up expressions like Cross[a,b] + Cross[b,c]. The Simplify command will perform almost all of the algebraic simplifications that are possible for a given input expression.] Post-experiment reading The result that i j = j i shouldn t be too surprising, since in the Preparatory Reading we spoke about the vector moment T = r F having a direction defined by turning an imaginary screwdriver from r to F. If instead we wanted to turn from F to r we must be describing the vector T = F r. There s another way to look at the results for i j etc. We know that the Cartesian basis (i, j, k) is a right-handed system. So when we seek the vector i j we re looking for a vector of magnitude 1 (a unit vector) in the direction which forms a right-handed system with i and j that s got to be k. If we consider now the permutations ( shufflings ) of (i, j, k), then just two of them are right-handed: (j, k, i) and (k, i, j). These imply a couple more cross product results. The other permutations, (j, i, k), (k, j, i) and (i, k, j), are not right-handed which tells us immediately that j i can t be the same as i j. General properties of the vector product: (i) the outcome of the vector product is always a vector; (ii) for any vector a, a a = 0; (iii) the vector product is not commutative, because for any a and b,

15 a b = b a We say that the vector product is anti-commutative, because the product one way around is the negative of the product the other way around. (iv) the vector product is distributive: a (b + c) = a b + b c for all vectors a, b and c. (v) the vector product is not associative: the relation (a b) c = a (b c) is not generally true. The fact that the vector product is distributive is really the thing which justifies our calling it a product in the first place. Experiment 6: The vector product in component form Preparatory reading In this experiment we seek a general formula for the vector product of vectors expressed in component form. The distributivity property found in the last experiment is key for this, because it means we can expand out the brackets in the general expression: (a x i + a y j+ a z k) (b x i + b y j+ b z k) where we re writing any vector a as a x i + a y j + a z k with a x denoting the x component of a, and so forth. 1) Perform by hand the expansion of the brackets in this general expression for a vector product: (a x i + a y j+ a z k) (b x i + b y j+ b z k) Use the results about i j etc. from Experiment 5 to simplify the expanded expression. Check your answer with Mathematica by doing, for example: Cross[Ax i + Ay j + Az k, Bx i + By j + Bz k] 2) If you have studied matrices and determinants, find out the relationship between a b and the following determinant: i j k a x a y a z. b x b y b z Compare your answer with the result from part 1. Post-experiment reading The general formula for the vector product in component form is: a b = (a y b z a z b y )i + (a z b x a x b z )j+(a x b y a y b x )k.

16 It turns out that the formula can also be written as a determinant of a matrix: a b = i j k a x a y a z. b x b y b z Practice Questions There is a set of questions for you to practice doing cross products in component form. To generate a question type: GiveQuestion["cross product"] not forgetting to "shift-return". To generate the answer for checking, type: LastAnswer["cross product"]

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. 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

More information

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.

More information

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

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

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

Section 1.1. Introduction to R n

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

More information

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

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

More information

Unified Lecture # 4 Vectors

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,

More information

Figure 1.1 Vector A and Vector F

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

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

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

More information

9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes

9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of

More information

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) 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

More information

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

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

More information

Solving Simultaneous Equations and Matrices

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

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

THREE DIMENSIONAL GEOMETRY

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,

More information

5.3 The Cross Product in R 3

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

More information

FURTHER VECTORS (MEI)

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

More information

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 1 NON-CONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects

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

Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi

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

More information

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors. 3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

More information

MATHS LEVEL DESCRIPTORS

MATHS LEVEL DESCRIPTORS MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and

More information

Vectors and Index Notation

Vectors and Index Notation Vectors and Index Notation Stephen R. Addison January 12, 2004 1 Basic Vector Review 1.1 Unit Vectors We will denote a unit vector with a superscript caret, thus â denotes a unit vector. â â = 1 If x is

More information

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 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

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

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

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

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

Vectors VECTOR PRODUCT. Graham S McDonald. A Tutorial Module for learning about the vector product of two vectors. Table of contents Begin Tutorial

Vectors VECTOR PRODUCT. Graham S McDonald. A Tutorial Module for learning about the vector product of two vectors. Table of contents Begin Tutorial Vectors VECTOR PRODUCT Graham S McDonald A Tutorial Module for learning about the vector product of two vectors Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk 1. Theory 2. Exercises

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

GCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook

GCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a

More information

Mathematics Notes for Class 12 chapter 10. Vector Algebra

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

More information

The Geometry of the Dot and Cross Products

The Geometry of the Dot and Cross Products Journal of Online Mathematics and Its Applications Volume 6. June 2006. Article ID 1156 The Geometry of the Dot and Cross Products Tevian Dray Corinne A. Manogue 1 Introduction Most students first learn

More information

5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1

5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1 5 VECTOR GEOMETRY Chapter 5 Vector Geometry Objectives After studying this chapter you should be able to find and use the vector equation of a straight line; be able to find the equation of a plane in

More information

State of Stress at Point

State of Stress at Point State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

More information

Vector Math Computer Graphics Scott D. Anderson

Vector Math Computer Graphics Scott D. Anderson Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about

More information

Vector Algebra II: Scalar and Vector Products

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

More information

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

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

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

Vectors Math 122 Calculus III D Joyce, Fall 2012

Vectors Math 122 Calculus III D Joyce, Fall 2012 Vectors Math 122 Calculus III D Joyce, Fall 2012 Vectors in the plane R 2. A vector v can be interpreted as an arro in the plane R 2 ith a certain length and a certain direction. The same vector can be

More information

Essential Mathematics for Computer Graphics fast

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

More information

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col

More information

Mechanics lecture 7 Moment of a force, torque, equilibrium of a body

Mechanics lecture 7 Moment of a force, torque, equilibrium of a body G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and

More information

The Geometry of the Dot and Cross Products

The Geometry of the Dot and Cross Products The Geometry of the Dot and Cross Products Tevian Dray Department of Mathematics Oregon State University Corvallis, OR 97331 tevian@math.oregonstate.edu Corinne A. Manogue Department of Physics Oregon

More information

Section 1: How will you be tested? This section will give you information about the different types of examination papers that are available.

Section 1: How will you be tested? This section will give you information about the different types of examination papers that are available. REVISION CHECKLIST for IGCSE Mathematics 0580 A guide for students How to use this guide This guide describes what topics and skills you need to know for your IGCSE Mathematics examination. It will help

More information

Part I. Basic Maths for Game Design

Part I. Basic Maths for Game Design Part I Basic Maths for Game Design 1 Chapter 1 Basic Vector Algebra 1.1 What's a vector? Why do you need it? A vector is a mathematical object used to represent some magnitudes. For example, temperature

More information

v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)

v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product) 0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3-space. This time the outcome will be a vector in 3-space. Definition

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

The Australian Curriculum Mathematics

The Australian Curriculum Mathematics The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year

More information

In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

More information

Section V.3: Dot Product

Section V.3: Dot Product Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here,

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

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

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

More information

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

More information

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

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

More information

Vector Algebra CHAPTER 13. Ü13.1. Basic Concepts

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

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS 9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

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

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

... ... . (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,

More information

2 Session Two - Complex Numbers and Vectors

2 Session Two - Complex Numbers and Vectors PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

More information

1 VECTOR SPACES AND SUBSPACES

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

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

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

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,

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

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

More information

CAMI Education linked to CAPS: Mathematics

CAMI Education linked to CAPS: Mathematics - 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to

More information

Math 215 HW #6 Solutions

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

More information

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

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

More information

WORK SCHEDULE: MATHEMATICS 2007

WORK SCHEDULE: MATHEMATICS 2007 , K WORK SCHEDULE: MATHEMATICS 00 GRADE MODULE TERM... LO NUMBERS, OPERATIONS AND RELATIONSHIPS able to recognise, represent numbers and their relationships, and to count, estimate, calculate and check

More information

Problem set on Cross Product

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

More information

The Dot and Cross Products

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

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

Mechanics 1: Vectors

Mechanics 1: Vectors Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized

More information

Lecture notes on linear algebra

Lecture notes on linear algebra Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra

More information

The Vector or Cross Product

The Vector or Cross Product The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero

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

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

More information

THEORETICAL MECHANICS

THEORETICAL MECHANICS PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents

More information

Standards and progression point examples

Standards and progression point examples Mathematics Progressing towards Foundation Progression Point 0.5 At 0.5, a student progressing towards the standard at Foundation may, for example: connect number names and numerals with sets of up to

More information

Common Core Unit Summary Grades 6 to 8

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

More information

DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x

DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of

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

Bedford, Fowler: Statics. Chapter 4: System of Forces and Moments, Examples via TK Solver

Bedford, Fowler: Statics. Chapter 4: System of Forces and Moments, Examples via TK Solver System of Forces and Moments Introduction The moment vector of a force vector,, with respect to a point has a magnitude equal to the product of the force magnitude, F, and the perpendicular distance from

More information

Module 8 Lesson 4: Applications of Vectors

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

More information

Math 1050 Khan Academy Extra Credit Algebra Assignment

Math 1050 Khan Academy Extra Credit Algebra Assignment Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Linear equations Ax D b come from steady state problems. Eigenvalues have their greatest importance in dynamic problems. The solution

More information

National 5 Mathematics Course Assessment Specification (C747 75)

National 5 Mathematics Course Assessment Specification (C747 75) National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for

More information

VECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a.

VECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a. VECTOR ALGEBRA Chapter 10 101 Overview 1011 A quantity that has magnitude as well as direction is called a vector 101 The unit vector in the direction of a a is given y a and is represented y a 101 Position

More information

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 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

More information

Numerical Analysis Lecture Notes

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

More information

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

More information

Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

More information

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA. Paul Dawkins COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

0 Introduction to Data Analysis Using an Excel Spreadsheet

0 Introduction to Data Analysis Using an Excel Spreadsheet Experiment 0 Introduction to Data Analysis Using an Excel Spreadsheet I. Purpose The purpose of this introductory lab is to teach you a few basic things about how to use an EXCEL 2010 spreadsheet to do

More information

Chapter 3 Vectors. m = m1 + m2 = 3 kg + 4 kg = 7 kg (3.1)

Chapter 3 Vectors. m = m1 + m2 = 3 kg + 4 kg = 7 kg (3.1) COROLLARY I. A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately. Isaac Newton - Principia

More information