11.1 Vectors in the plane

Transcription

1 11.1 Vectors in the plane What is a vector? It is an object having direction and length. Geometric way to represent vectors It is represented by an arrow. The direction of the arrow is the direction of the vector, the length of the arrow is the length of the vector. 1 / 17

2 a b c d Exercise. Why do we say that vectors have no location? Sketch two vectors to illustrate what you mean. If A and B are two points in the plane, sketch AB. With respect to the vector AB, what do we call A and B? Sketch a vector equal to AB in which the initial point is the origin. Can this always be done? 2 / 17

3 Representing Vectors Mathematically Let P = (x, y) be a point in the plane, with P not located at the origin. If v is the vector from the origin to P, we denote it in any of the following ways: v = OP =< x, y > We ll see another way to represent a vector later on in the section. When we sketch a vector in which the initial point is the origin, we say we are sketching it as a position vector. When we use the notation < x, y > to denote a vector v, we call it the position vector representation of v. Note that if we aren t using the position vector representation, in order to communicate clearly, we must write an arrow over the symbol to indicate it is a vector. 3 / 17

4 Exercise. a What is the difference between the notation (2, 3) and the notation < 2, 3 >? b Sketch the vector PQ from P = (1, 3) to Q = (5, 12). c What is the position vector representation of PQ? Illustrate it by sketching it as a position vector. 4 / 17

5 Arithmetic with Vectors Some Arithmetic with Vectors We use the position vector representation < x, y > to define the various rules of arithmetic of vectors. 1 Components of a vector: If v =< 2, 5 >, we call 2 the x-component of v and we call 5 the y-component of v. 2 The 0-vector: 0 =< 0, 0 > 3 Equality of vectors: < x 1, y 1 >=< x 2, y 2 > if and only if x 1 = x 2 and y 1 = y 2. Exercise. If v is represented in the cryptic way v =< 2x + y, 4x y >, but we also know that v =< 5, 7 >, what do the above rules allow us to say about x and y? 5 / 17

6 Arithmetic with Vectors Scalar Multiplication The term scalar refers to a real number. Scalar multiplication refers to the operation of multiplying a vector by a real number. If c is a real number and < x, y > is a vector, then we define c < x, y >:=< c x, c y > Note that the multiplication of a vector by a scalar produces a vector, not a scalar. Exercise. 1 Let v =< 3, 4 >. a How much is 2 v? How much is 2 v? b For this example, plot v, 2 v, and 2 v in the position vector representation. 2 For a generic vector v, illustrate what v, 2 v, and 2 v look like. 6 / 17

7 Arithmetic with Vectors Parallel Vectors Two vectors v and w are called parallel if they point either in the same or opposite direction. The sketches on the previous page make the following clear: v and w are parallel if and only if there exists a nonzero scalar c such that v = c w Exercise. If v =< 6, 9 > and w =< 12, t >, is there any choice of t for which v and w are parallel? If so, which value of t? 7 / 17

8 Arithmetic with Vectors Vector Addition If < x 1, y 1 > and < x 2, y 2 > are two vectors, we define their vector sum by: < x 1, y 1 > + < x 2, y 2 >:=< x 1 + x 2, y 1 + y 2 > Note that the vector sum of two vectors is a vector, not a scalar. Exercise. 1 If v =< 1, 1 > and w =< 1, 2 >, calculate v + w. Then sketch all three of them on the same plot as position vectors. What do you notice? 2 Explain why we say that vector addition satisfies the parallelogram rule. Illustrate what it says for generic vectors v and w. 3 Given a parallelogram with sides v and w, identify the two diagonals in terms of linear combinations of v and w. 8 / 17

9 Arithmetic with Vectors R 2 as a Vector Space We will sometimes view R 2 as being the set of all ordered pairs of real numbers (x, y), and we will other times view R 2 as being the set of two-dimensional vectors < x, y >. When we view R 2 as being the set of two-dimensional vectors, we refer to the operations of scalar multiplication and vector addition as being the vector space operations, and we refer to R 2 as being a vector space. One can build entire courses around the idea of viewing R 2 (and generalizations of R 2 ) as a vector space. That is what is done in Math 203 and Math / 17

10 Norm Structure of R 2 Recall we said that any vector has both a direction and a magnitude. The terms magnitude, length, and norm all mean the same thing. They refer to the length of the arrow representing the vector. Calculation of the magnitude of a vector If v =< x, y >, then the magnitude of v is denoted by v and is defined to be v := x 2 + y 2 Notice that v is a scalar quantity, not a vector. Notice that the only way that v can be 0 is if both x and y are 0: v = 0 if and only if v = 0 Also notice the effect on magnitude upon multiplication by a scalar: c v = c < x, y >= < cx, cy > = c 2 x 2 + c 2 y 2 = c 2 (x 2 + y 2 ) = c x 2 + y 2 = c v 10 / 17

11 Norm Structure of R 2 Exercise. 1 a Calculate < 1, 1 > and < 1, 1 >. Illustrate your answer by drawing an appropriate sketch. b Calculate also < 1, 1 > + < 1, 1 >. What is the relation to < 1, 1 > + < 1, 1 >? 2 More generally, explain geometrically why it is true that v + w v + w Explain why this inequality is referred to as the triangle inequality. Is it ever possible to have vectors v and w for which we get equality in the triangle inequality? 11 / 17

12 Norm Structure of R 2 So to recap, the properties of norm are as follows: Properties of norm 1 v = 0 if and only if v = 0 2 c v = c v 3 (triangle inequality) v + w v + w with equality if and only if v and w point in exactly the same direction. When we view R 2 with its vector space structure and its norm structure, we refer to it as an example of a normed space. 12 / 17

13 Norm Structure of R 2 We say v is a unit vector if v = 1. Unit vectors and vectors of prescribed length Given any vector v such that v 0, there is a unique unit vector in the same direction as v. It is given by 1 v v If c is a posiitve scalar, then the vector c v points in the same direction as v but has c times the length of v. On the other hand, to get a vector pointing in the same direction as v but which has length c we do instead 1 c v. v 13 / 17

14 Exercise. a Explain why the above formula works for producing a unit vector in the direction of the vector v. b Write down the vector in the same direction as v but which has length 7 times the length of v. c Say instead you want a vector in the same direction as v but which has length 7, how would you do it? d For the vector v =< 3, 5 >, write down a vector in the opposite direction to v which has length / 17

15 Coordinate Unit Vectors ı and j It s convenient to have special notations for the unit vectors in the direction of the positive coordinate axes. So we define ı :=< 1, 0 > and j :=< 0, 1 > a b c d Exercise. Sketch ı and j as position vectors. Show how to write the vector < 15, 3 > in terms of ı and j. Show how to write the generic vector < x, y > in terms of ı and j. Write the vector joining P = (15, 3) to Q = (11, 20) in terms of ı and j. 15 / 17

16 Linear Combinations of Vectors If v and w are any vectors and c 1, c 2 any scalars, we call the expression a linear combination of v and w. c 1 v + c2 w So for example, when we write < 2, 5 > in the equivalent way as 2 ı + 5 j, we are writing < 2, 5 > as a linear combination of ı and j. As another example, if v =< 1, 1 > and w =< 1, 1 >, then 5 v + w =< 6, 4 >, so in writing < 6, 4 >= 5 < 1, 1 > + < 1, 1 >, we are writing < 6, 4 > as a linear combination of < 1, 1 > and < 1, 1 >. 16 / 17

17 Exercise. 1 If < x, y > is a generic vector, show that it can be written as a linear combination of < 1, 1 > and < 1, 1 >. 2 Give a geometric argument which generalizes the result of number 1. That is, if v and w are any two vectors which are not parallel, give a geometric argument which explains why any generic vector can be written as a linear combination of v and w. 17 / 17