Let a = i + j - k , b = i - j , and c = 2 i + j + 2 k . A vector v satisfies v a = b and v c = 3 . If v is expressed as v = v ₁ + v ₂ , where v ₁ is parallel to c and v ₂ is perpendicular to c , then the value of | v ₂|^2 is equal to

Step-by-step solution

Given the vector equation v a = b , take the cross product with c on both sides: c ( v a ) = c b Using the vector triple product expansion: ( c a ) v - ( c v ) a = c b First, calculate the required dot products: c a = (2)(1) + (1)(1) + (2)(-1) = 2 + 1 - 2 = 1 We are given c v = 3 . Substitute these into the expanded equation: (1) v - 3 a = c b v = c b + 3 a Now, compute the cross product c b : c b = (2 i + j + 2 k ) ( i - j ) = i (0 - (-2)) - j (0 - 2) + k (-2 - 1) = 2 i + 2 j - 3 k Compute 3 a : 3 a = 3( i + j - k