Let a vector c⃗ be coplanar with the vectors a⃗ = - î + ĵ + k̂ and b⃗ = 2î + ĵ - k̂. If the vector c⃗ also satisfies the conditions c⃗ . [ ( a⃗ + b⃗ ) × ( a⃗ × b⃗ ) ] = - 42 and ( c⃗ × ( a⃗ - b⃗ ) ) . k̂ = 3, then the value of |c⃗ |² is equal to :

Solution

Maths sir · 4 steps

Coplanar means c⃗ = xa⃗ + yb⃗, and each condition gives one equation in x and y.

1. Coplanar form. Since c⃗ lies in the plane of a⃗ and b⃗, take c⃗ = xa⃗ + yb⃗. Keep the dot products ready: a⃗ · a⃗ = 3, b⃗ · b⃗ = 6, a⃗ · b⃗ = -2 + 1 - 1 = -2.

   (a⃗ + b⃗) × (a⃗ × b⃗) = ((a⃗ + b⃗)· b⃗)a⃗ - ((a⃗ + b⃗)· a⃗)b⃗ = 4a⃗ - b⃗

2. First condition. Here (a⃗ + b⃗)· b⃗ = -2 + 6 = 4 and (a⃗ + b⃗)· a⃗ = 3 - 2 = 1. With c⃗ · a⃗ = 3x - 2y and c⃗ · b⃗ = -2x + 6y, the condition 4 c⃗ · a⃗ - c⃗ · b⃗ = -42 gives 14x - 14y = -42, so x - y = -3.
3. Second condition. c⃗ × (a⃗ - b⃗) = (xa⃗ + yb⃗) × (a⃗ - b⃗) = -(x + y) a⃗ × b⃗, and a⃗ × b⃗ = -2î + ĵ - 3k̂. Its k̂ part gives 3(x + y) = 3, so x + y = 1.
4. So. x = -1 and y = 2, so c⃗ = -a⃗ + 2b⃗ = 5î + ĵ - 3k̂. Then |c⃗|² = 25 + 1 + 9 = 35, option (c).

The idea to keep. Coplanar with a⃗ and b⃗ means xa⃗ + yb⃗. Then open the triple product with p⃗ × (q⃗ × r⃗) = (p⃗ · r⃗)q⃗ - (p⃗ · q⃗)r⃗.

Answer from the NTA final answer key. Solution by PrepWiser.