Let a⃗, b⃗ and c⃗ be three non zero vectors such that b⃗ . c⃗ = 0 and a⃗ × (b⃗ × c⃗ ) = (b⃗ - c⃗)/2. If d⃗ be a vector such that b⃗ . d⃗ = a⃗ . b⃗ , then (a⃗ × b⃗ ) . (c⃗ × d⃗ ) is equal to

Solution

Maths sir · 4 steps

Expand the triple product, match the coefficients of b and c, and both needed dot products drop out.

1. Expand the triple product. Use a⃗ × (b⃗ × c⃗) = (a⃗ · c⃗) b⃗ - (a⃗ · b⃗) c⃗. This must equal 1/2b⃗ - 1/2c⃗.
2. Match coefficients. b⃗ and c⃗ are non-zero and perpendicular, so neither is a multiple of the other. Their coefficients must match on both sides, which gives a⃗ · c⃗ = 1/2 and a⃗ · b⃗ = 1/2.

   (a⃗ × b⃗) · (c⃗ × d⃗) = (a⃗ · c⃗)(b⃗ · d⃗) - (a⃗ · d⃗)(b⃗ · c⃗)

3. Put in values. The second term is zero because b⃗ · c⃗ = 0. Also b⃗ · d⃗ = a⃗ · b⃗ = 1/2. The first term is 1/2 × 1/2 = 1/4.
4. So. (a⃗ × b⃗) · (c⃗ × d⃗) = 1/4, which is option (c).

The idea to keep. A dot product of two cross products always becomes a 2 by 2 determinant of dot products (Lagrange identity). Find those dot products first, often by comparing coefficients of independent vectors.

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