SECTION 9.4 Vectors 639 For example, if a is the acceleration of an object of mass m due to a force F being exerted on it, then, by Newton’s second law of motion, = m F a. Here, ma is the product of the scalar m and the vector a . Scalar multiples have the following properties: • = v 0 0 = v v • 1 − = − v v • 1 • α β α β ( ) + = + v v v α α α ( ) + = + v w v w • • α β αβ ( ) ( ) = v v 1 Graph Vectors Graphing Vectors Use the vectors illustrated in Figure 54 to graph each of the following vectors: (a) −v w (b) +v w 2 3 (c) − + v w u 2 Solution EXAMPLE 1 Figure 55 shows each graph. Figure 55 v 2 w (a) v 2 w 2v 1 3w (b) 2v 1 3w 2v 2 w 1 u (c) 2v 2 w 1 u 2 w v 2v 3w u 2 w 2v Figure 54 w u v Now Work PROBLEMS 11 AND 13 Magnitude of Vectors The symbol v represents the magnitude of a vector v . Since v equals the length of a directed line segment, it follows that v has the following properties: THEOREM Properties of the Magnitude v of a Vector v If v is a vector and if α is a scalar, then (a) ≥ v 0 (b) = v 0 if and only if = v 0 (c) − = v v (d) α α = v v Property (a) is a consequence of the fact that distance is a nonnegative number. Property (b) follows because the length of the directed line segment PQ is positive unless P and Q are the same point, in which case the length is 0. Property (c) follows because the length of the line segment PQ equals the length of the line segment QP. Property (d) is a direct consequence of the definition of a scalar multiple. DEFINITION Unit Vector A vector u for which = u 1 is called a unit vector .
RkJQdWJsaXNoZXIy NjM5ODQ=