SECTION 9.6 Vectors in Space 663 Recall that a unit vector u is one for which = u 1. In many applications, it is useful to be able to find a unit vector u that has the same direction as a given vector v . THEOREM Unit Vector in the Direction of v For any nonzero vector v , the vector u v v = is a unit vector that has the same direction as v . The following is a consequence of this theorem. v v u = Finding a Unit Vector Find the unit vector in the same direction as v i j k 2 3 6 . = − − Solution EXAMPLE 5 Find v first. v i j k 2 3 6 4 9 36 49 7 = − − = + + = = Now multiply v by the scalar v 1 1 7 . = The result is the unit vector u v v i j k i j k 2 3 6 7 2 7 3 7 6 7 = = − − = − − Now Work PROBLEM 47 4 Find the Dot Product The definition of dot product in space is an extension of the definition given for vectors in a plane. DEFINITION Dot Product If a b c v i j k 1 1 1 = + + and a b c w i j k 2 2 2 = + + are two vectors, the dot product v w⋅ is defined as a a b b c c v w 1 2 1 2 1 2 ⋅ = + + (3) Finding Dot Products If v i j k 2 3 6 = − + and w i j k 5 3 , = + − find: (a) v w⋅ (b) w v ⋅ (c) v v ⋅ (d) w w⋅ (e) v (f) w Solution EXAMPLE 6 (a) v w 2 5 3 3 6 1 5 ( ) ( ) ⋅ = ⋅ + − + − = − (b) w v 5 2 3 3 1 6 5 ( ) ( ) ⋅=⋅+−+−⋅=− (c) v v 2 2 3 3 6 6 49 ( )( ) ⋅ = ⋅ + − − + ⋅ = (d) w w 5 5 3 3 1 1 35 ( )( ) ⋅ = ⋅ + ⋅ + − − = (e) v 2 3 6 49 7 2 2 2 ( ) = + − + = = (f) w 5 3 1 35 2 2 2 ( ) = + + − =
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