If a⃗ = î + 2k̂,b⃗ = î + ĵ + k̂,c⃗ = 7î - 3ĵ + 4k̂,r⃗ × b⃗ + b⃗ × c⃗ = 0⃗ and r⃗ . a⃗ = 0. Then r⃗ . c⃗ is equal to :

Solution

Maths sir · 4 steps

Rewrite b⃗ × c⃗ as -c⃗ × b⃗, and both cross products collapse into one.

1. Combine the terms. Since b⃗ × c⃗ = -c⃗ × b⃗, the condition becomes r⃗ × b⃗ - c⃗ × b⃗ = 0⃗, that is (r⃗ - c⃗) × b⃗ = 0⃗.
2. Parallel vectors. A zero cross product means r⃗ - c⃗ is parallel to b⃗. So r⃗ = c⃗ + λ b⃗ for some scalar λ.
3. Find lambda. Use r⃗ · a⃗ = 0. Here c⃗ · a⃗ = 7 + 0 + 8 = 15 and b⃗ · a⃗ = 1 + 0 + 2 = 3, so 15 + 3λ = 0 and λ = -5.

   r⃗ · c⃗ = c⃗ · c⃗ - 5 (b⃗ · c⃗) = 74 - 5 × 8

4. So. With c⃗ · c⃗ = 49 + 9 + 16 = 74 and b⃗ · c⃗ = 7 - 3 + 4 = 8, we get r⃗ · c⃗ = 74 - 40 = 34, option (c). As a check, r⃗ = 2î - 8ĵ - k̂ gives 14 + 24 - 4 = 34.

The idea to keep. Whenever you see r⃗ × b⃗ = c⃗ × b⃗, read it as r⃗ = c⃗ + λ b⃗. Then one dot-product condition fixes λ for you.

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