SECTION 9.4 Vectors 643 In Words Dividing a vector v by its magnitude transforms v into a vector whose length equals 1. THEOREM Unit Vector in the Direction of v For any nonzero vector v , the vector = u v v (6) is a unit vector that has the same direction as v . If u is a unit vector that has the same direction as a vector v , then v can be expressed as = v v u (7) Proof Let = + a b v i j. Then = + a b v 2 2 and = = + + = + + + a b a b a a b b a b u v v i j i j 2 2 2 2 2 2 The vector u has the same direction as v , since > v 0. Also, = + + + = + + = a a b b a b a b a b u 1 2 2 2 2 2 2 2 2 2 2 That is, u is a unit vector that has the same direction as v . ■ The following is a consequence of this theorem. Finding a Unit Vector Find a unit vector that has the same direction as = − v i j 4 3 . Solution EXAMPLE 5 Find v first. = − = + = v i j 4 3 16 9 5 Now multiply v by the scalar = v 1 1 5 . A unit vector that has the same direction as v is = − = − v v i j i j 4 3 5 4 5 3 5 Check: This vector is a unit vector because ( ) ( ) = + − = + = = v v 4 5 3 5 16 25 9 25 25 25 1 2 2 Now Work PROBLEM 53 6 Find a Vector from Its Direction and Magnitude If a vector represents the speed and direction of an object, it is called a velocity vector . If a vector represents the direction and amount of a force acting on an object, it is called a force vector . In many applications, a vector is described in terms of its
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