Let a → = 2 i ^ - j ^ + 2 k ^ and b → = i ^ + 2 j ^ - k ^ . Let a vector v → be in the plane containing a → and b → . If v → is perpendicular to the vector 3 i ^ + 2 j ^ - k ^ and its projection on a → is 19 units, then | 2 v → | 2 is equal to _____.

Step-by-step solution

v → , a → , b → are coplanar v → = λ a → + μ b → v → = 2 λ + μ i ^ + 2 μ - λ j ^ + 2 λ - μ k ^ given v → ⊥ 3 i ^ + 2 j ^ - k ^ v → · 3 i ^ + 2 j ^ - k ^ = 0 ⇒ ( 2 λ + μ i ^ + 2 μ - λ j ^ + 2 λ - μ k ^ ) . ( 3 i ^ + 2 j ^ - k ^ ) = 0 ⇒ 3 ( 2 λ + μ ) + 2 ( 2 μ - λ ) - ( 2 λ - μ ) = 0 ⇒ λ = - 4 μ       . . . ( i )