Let c⃗ be a vector perpendicular to the vectors, a⃗ = î + ĵ - k̂ and
b⃗ = î + 2ĵ + k̂. If c⃗ . ( î + ĵ + 3k̂ ) = 8 then the value of
c⃗ . ( a⃗ × b⃗ ) is equal to __________.

Solution

Maths sir · 3 steps

A vector perpendicular to both a and b must lie along a × b, so find that first.

1. Direction of c. Since c⃗ is perpendicular to both a⃗ and b⃗, it is parallel to a⃗ × b⃗. So write c⃗ = λ(a⃗ × b⃗) for some scalar λ.

   a⃗ × b⃗ = |î, ĵ, k̂; 1, 1, -1; 1, 2, 1| = 3î - 2ĵ + k̂

2. Find lambda. c⃗ · (î + ĵ + 3k̂) = λ(3 - 2 + 3) = 4λ = 8, so λ = 2 and c⃗ = 6î - 4ĵ + 2k̂.
3. So. c⃗ · (a⃗ × b⃗) = λ|a⃗ × b⃗|² = 2 × (9 + 4 + 1) = 2 × 14 = 28.

The common slip: 14. Forgetting the minus sign on the ĵ term of the determinant gives a⃗ × b⃗ = 3î + 2ĵ + k̂. Then λ = 1 and the answer comes out as 14.

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

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