Let a⃗=î+2 ĵ+k̂,
b⃗=3(î-ĵ+k̂).
Let c⃗ be the vector such that a⃗ × c⃗=b⃗ and a⃗ · c⃗=3.
Then a⃗ ·((c⃗ × b⃗)-b⃗-c⃗) is equal to :

Solution

Maths sir · 4 steps

Swap the dot and cross in the triple product, and you never need to find c⃗ itself.

1. Split the expression. The dot product distributes over the bracket, so the expression is a⃗ · (c⃗ × b⃗) - a⃗ · b⃗ - a⃗ · c⃗. Find each of the three pieces.
2. Rotate the triple product. A scalar triple product does not change when you swap the dot and the cross. So a⃗ · (c⃗ × b⃗) = (a⃗ × c⃗) · b⃗ = b⃗ · b⃗.

   b⃗ · b⃗ = 3² + (-3)² + 3² = 27

3. The other two. a⃗ · b⃗ = 3(1 - 2 + 1) = 0. This must be zero anyway, because b⃗ = a⃗ × c⃗ is perpendicular to a⃗. Also, a⃗ · c⃗ = 3 is given.
4. So. The value is 27 - 0 - 3 = 24, which is option (c).

The idea to keep. If a cross product is given and the unknown vector sits inside a triple product, swap the dot and the cross. The given value drops straight in, so you never solve for the vector.

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