638 CHAPTER 9 Polar Coordinates; Vectors Adding Vectors Geometrically The sum +v w of two vectors is defined as follows: Position the vectors v and w so that the terminal point of v coincides with the initial point of w , as shown in Figure 48. The vector +v w is then the unique vector whose initial point coincides with the initial point of v and whose terminal point coincides with the terminal point of w . Figure 48 Adding vectors v 1 w Initial point of v Terminal point of w w v Figure 50 ( ) ( ) + + = + + u v w u v w w u 1 v v 1 w u v Figure 49 + = + v w w v v 1 w w 1 v w v v w Figure 51 Opposite vectors v 2 v Figure 52 ( ) − = + − v w v w 2 w v v 2 w Figure 53 Scalar multiples 21v 2v v Vector addition is commutative . That is, if v and w are any two vectors, then + = + v w w v Vector addition is also associative . That is, if u , v , and w are vectors, then ( ) ( ) + + = + + u v w u v w The zero vector 0 has the property that + = + = v 0 0 v v for any vector v . ( ) + − = v v 0 Figure 49 illustrates this fact. (Observe that the commutative property is another way of saying that opposite sides of a parallelogram are equal and parallel.) Figure 50 illustrates the associative property for vectors. If v is a vector, then −v is the vector that has the same magnitude as v , but whose direction is opposite to v , as shown in Figure 51. Furthermore, If v and w are two vectors, then the difference −v w is defined as ( ) − = + − v w v w Figure 52 illustrates the relationships among v w , , and −v w. Multiplying Vectors by Numbers Geometrically When using vectors, real numbers are referred to as scalars . Scalars are quantities that have only magnitude. Examples of scalar quantities from physics are temperature, speed, and time. We now define how to multiply a vector by a scalar. DEFINITION Scalar Multiple If α is a scalar and v is a vector, the scalar multiple αv is defined as follows: • If α α > v 0, is the vector whose magnitude is α times the magnitude of v and whose direction is the same as that of v . • If α α < v 0, is the vector whose magnitude is α times the magnitude of v and whose direction is opposite that of v . • If α = 0 or if = v 0, then α = v 0. See Figure 53 for some illustrations.
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