Let a = i + j + k , b = 2 i - j + k and c = i + 2 j - k . Let v be a vector in the plane of a and b such that v is perpendicular to a . If the length of the projection of v on c is 14 6 , then | v |^2 is equal to

Step-by-step solution

Let v = a + b . Since v is perpendicular to a , we have v a = 0 . ( a + b ) a = 0 | a |^2 + ( a b ) = 0 . Here, | a |^2 = 1^2 + 1^2 + 1^2 = 3 and a b = (1)(2) + (1)(-1) + (1)(1) = 2 . Thus, 3 + 2 = 0 = 2k and = -3k for some scalar k . Substituting these, v = 2k( i + j + k ) - 3k(2 i - j + k ) = k(-4 i + 5 j - k ) . The projection of v on c is given by | v c | | c | . We have v c = k(-4(1) + 5(2) - 1(-1)) = 7k . Also, | c | = 1^2 + 2^2 + (-1)^2 = 6 . Given that the projection length is 14 6 , we get |7k| 6 = 14 6 |k