Let a be a vector such that its projections on the vectors i + j , j + k , and k + i are 2 2 , 3 2 , and 3 2 respectively. Let b be a vector such that | b | = | a | and a b = 0 . If a , b , and i + k are coplanar, then the magnitude of the projection of vector b on the vector i - 2 j + 2 k is equal to

Step-by-step solution

Let a = a₁ i + a₂ j + a₃ k . Given the projections of a on i + j , j + k , and k + i : a₁+a₂ 2 = 2 2 a₁+a₂ = 4 a₂+a₃ 2 = 3 2 a₂+a₃ = 3 a₃+a₁ 2 = 3 2 a₃+a₁ = 3 Adding these three equations gives 2(a₁+a₂+a₃) = 10 a₁+a₂+a₃ = 5 . Solving for the components, we get a₁ = 2 , a₂ = 2 , and a₃ = 1 . So, a = 2 i + 2 j + k and | a | = 4+4+1 = 3 . Since a , b , and i + k are coplanar, we can write: b = x a + y( i + k ) Given a b = 0 : a (x a + y( i + k )) = 0 x| a |^2 + y( a ( i + k )) = 0 x(9) + y(2(1) + 2(0) + 1(1)) = 0 9x +