Let a⃗=2 î+7 ĵ-k̂, b⃗=3 î+5 k̂ and c⃗=î-ĵ+2 k̂. Let d⃗ be a vector which is perpendicular to both a⃗ and b⃗, and c⃗ · d⃗=12. Then (-î+ĵ-k̂) ·(c⃗ × d⃗) is equal to :

Solution

Maths sir · 4 steps

A vector perpendicular to both a and b lies along a × b. Fix its length with c · d = 12, then compute.

1. Direction of d. A vector perpendicular to both a⃗ and b⃗ is parallel to a⃗ × b⃗. So d⃗ = λ(a⃗ × b⃗) for some number λ.

   a⃗ × b⃗ = |î, ĵ, k̂; 2, 7, -1; 3, 0, 5| = 35î - 13ĵ - 21k̂

2. Find lambda. c⃗ · d⃗ = λ(35 + 13 - 42) = 6λ = 12, so λ = 2. This gives d⃗ = 70î - 26ĵ - 42k̂.
3. Cross c with d. c⃗ × d⃗ has components ((-1)(-42) - 2(-26), 2(70) - 1(-42), 1(-26) - (-1)(70)) = (94, 182, 44).
4. So. (-î+ĵ-k̂) · (c⃗ × d⃗) = -94 + 182 - 44 = 44. The answer is (c).

The idea to keep. Perpendicular to two given vectors means parallel to their cross product. Write it as λ(a⃗ × b⃗) and let the extra condition fix λ.

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