Scalar multiplication and algebraic direction of a vector

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1 Roberto s Notes on Linear Algebra Chapter 1: Geometric ectors Section 5 Scalar multiplication and algebraic direction of a ector What you need to know already: of a geometric ectors. Length and geometric direction of a ector. What you can learn here: The operation of scalar multiplication. How to use this operation to construct an efficient definition of direction for a ector. According to our definitions, a 2D ector is simply an ordered pair of usual numbers and a 3D ector is an ordered triple of numbers. Although in each case we can deise and use a geometric representation, the algebraic aspect is the key feature that will be exploited in the rest of the course. As a way of introduction, I will now show you how, by using the arithmetic operations that are so familiar to us from grade school, we can deelop a useful operation for ectors that will lead to a much better way to identify a direction. The scalar multiplication of a scalar c and a ector is performed by multiplying each of the components of by c. Therefore: In 2D: c c c c In 3D: c c c c c Two ectors u and are scalar multiples of each other if there is a scalar c such that either c or u c. Example: By this definition: and It turns out that this is an extremely useful operation, one that forms the backbone for most of the material of this book. To start showing you how useful this is, here is the better definition of direction that I hae promised: u Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 1

2 Proof Technical fact Two non-zero geometric ectors hae the same direction if and only if they are scalar multiples of each other. Moreoer, they hae the same orientation if such scalars are positie. Same direction Scalar multiples: In 2D, if two ectors and w w 1 2 w hae the same 1 2 direction, it means that either they are both ertical, or they hae the same slope. If they are both ertical, they are of the form 0 2 and w 0 w 2 with both 2 and w 2 different from 0. In that case, if we w2 pick c we see that the two ectors are scalar multiples of each 2 other. Scalar multiples Same direction: If, w w 1 2 w and w c, it means that 1 2 w2 c2 2 w c1 c2 and hence, so that and w hae the w c same direction. This argument only fails if 1 0, but in that case w1 0 as well and hence both ectors are ertical and so they hae the same direction. I leae to you the proof for the 3D case. You want me to proe it in the more difficult situation? That s rich! And why do I need to be able to do proofs anyway? I trust what you say I am honored by your trust, but learning how to do simple proofs is one of the goals of this course, as well as a skill that will be ery useful to you in the future. So we all expect you to learn how to do it. As for the difficulty leel, in 3D the proof is the same, just longer, since there are more proportions to juggle, but the procedure is the same, so you ll get a good workout. In any case, now that we hae a more workable definition of direction, we can look at another word with which you are familiar and that we can now define formally: Two geometric ectors are parallel if they hae the same direction, or, equialently, if they are scalar multiples of each other. Example: If we consider the ectors 6 4 b 96, we a and 2 notice that b 1.5a, or a b. 3 This means that they are multiples of each other and hence they hae the same direction, or are parallel, as the picture clearly shows. a b Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 2

3 It looks like two ectors are parallel if one of them is a stretched ersion of the other. That is correct, since we are maintaining the direction. But there is more! If you look at the picture you may get the impression that b is about 1.5 times longer than a. This may seem obious, but remember that the scalar multiplication of the definition occurs on the components, not the length, so we need to check if this obseration is in fact true and not a coincidence: Proof Technical fact Scalar multiplication has the effect of changing the length of the ector by a factor equal to the absolute alue of the scalar factor: c Here is how to proe this fact in 2D: c c c c c c c c c c c 1 2 c Again I leae you the fun of proing this in 3. Yes, and in fact this gies us a ery useful way to obtain a unit ector in any direction we want. Technical fact Gien any non-zero ector, the ector 1 denoted by e, it is a unit ector and has the same direction and orientation as. The process of constructing a unit ector in the direction of a gien non-zero ector is called normalization. Example: If we normalize the ector 3 2 obtain: e Both ectors are shown in the picture and so is the ector 1 e 3 2, which has the same direction, but opposite 13 orientation we e e is So, we can use scalar multiplication to lengthen and shorten ectors at will, correct? I hae a question about jargon and one about notation. Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 3

4 Excellent! Both are ery important aspects of this course, so it is best to clarify them as soon as they arise. First of all, we called this operation scalar multiplication. Why not the shorter scalar product? It is fine to refer to this operation as the scalar product of a scalar and a ector, but later we shall meet another operation that is also called scalar product, but refers to a rather different situation and produces a different result. So, we can use this terminology as long as the context is clear. Fair enough. Now, when we multiply a ector by a fraction, can we make the notation smaller by placing the ector in the numerator? Absolutely! This is a ery conenient moe and we might as well formalize it in a box. A scalar diision of a ector by a scalar is obtained by multiplying the ector by the reciprocal of the scalar. That is: 1 c c When defining what we mean by two ectors being multiples of each other, I excluded consideration of the zero ector, for otherwise we may hae generated a diision by 0 in the formula. That is also reasonable because the zero ector does not hae a proper direction (it includes a single point) and because the constant that would change any ector to the zero ector is 0, but one cannot diide by zero in order to reerse the relation. Although reasonable, it turns out that this exclusion would gie us lots of headaches in future deelopments. To aoid this problem, we shall use the following conention: The zero ector is considered to be a multiple of any other ector, hence parallel to any other ector, despite the fact that it does not hae a direction and that the algebraic relationship of being scalar multiple would only be satisfied in one direction. I am delaying examples of this definition because at this point their simplicity would offend your intelligence and because we shall hae many opportunities to see it in action when it counts. For now, just keep it in mind and write down any questions you may hae about it. Before we close this chapter and moe to bigger and better things, I want to clarify a little pedantic point that may bother us later if we don t. Summary Since scalar multiplication changes each component by the same factor, it does not affect the direction of a ector. Therefore, it can be used to identify such direction through an equialence relation. Two ectors hae the same direction, or are parallel, if they are multiples of each other. Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 4

5 A unit ector can be seen as being representatie of a direction. Common errors to aoid Distinguish between direction and orientation: they are related concepts, but not the same thing! Remember that the zero ector is considered as being parallel to any other ector, een though this slightly iolates the mutual requirement of the definition. Learning questions for Section LA 1-5 Reiew questions: 1. Describe how scalar multiplication works. 3. Discuss the role of unit ectors in the concept of algebraic direction of a ector. 2. Explain how scalar multiplication is used to define the direction of a geometric ector. Memory questions: 1. Which formula defines the product of a scalar and a 3D ector? 2. Which formula defines the unit ector with the same direction and orientation of a gien ector? Computation questions: 1. Gien the ectors 3 5 and u 5.2 3, determine 3 and -2u. 2. Determine the unit ectors parallel to 3 5 and u Which ector is the unit ector in the direction of the ector PQ joining P 3, 2 1, 2? and Q Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 5

6 4. Determine the unit ector of that has the opposite orientation of the ector Determine the magnitude and the direction as a unit ector of the ector whose tail is at the point 3,1, 0 and whose tip is at 1, 2, Find all alues of k for which the ectors k k 2, 3 k 1 parallel. 7. Gien the ectors u 12kand u are, determine the alues of k for which, respectiely: a) u and are parallel and with the same orientation; b) u and are parallel and with opposite orientation. Theory questions: 1. How is a unit ector obtained from a gien ector? 2. What is the main purpose of unit ectors? 3. What is the geometric effect of multiplying a ector by a scalar? 4. Gien a ector abc, construct a ector w that has half its length. 5. How many unit ectors hae the same direction as a gien ector? 6. Which are the unit ectors in the direction of the line y x? Proof questions: 1. Show that if two 2D ectors a1 a 2 and 1 2 a 1 b1 0, then they hae the same orientation. bb hae the same direction and 2. Proe that scalar multiplication is distributie, that is, for any geometric ectors u and and any scalars h and k, h k u hu ku. h u hu h and 3. Proe that scalar multiplication is associatie, that is, for any geometric ector u h ku hk u. and scalars h and k, 4. Proe that the scalar 1 is an identity for scalar multiplication, that is, for any geometric ector, 1. Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 6

7 Templated questions: In these questions, make your own choice of any items inoled. 1. Choose a 2D ector and then construct the unit ector parallel to it and with the same orientation. 2. Choose a 3D ector and then construct the unit ector parallel to it and with the same orientation. 3. Choose a 2D ector and then construct the unit ector parallel to it and with the opposite orientation. 4. Choose a 3D ector and then construct the unit ector parallel to it and with the opposite orientation. What questions do you hae for your instructor? Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 7

8 Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 8

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