Let a⃗=î-3 ĵ+7 k̂, b⃗=2 î-ĵ+k̂ and c⃗ be a vector such that (a⃗+2 b⃗) × c⃗=3(c⃗ × a⃗). If a⃗ · c⃗=130, then b⃗ · c⃗ is equal to __________.

Solution

Maths sir · 4 steps

Move everything to one side: a cross product equal to zero means the two vectors are parallel.

1. Flip the order. Use c⃗ × a⃗ = -a⃗ × c⃗. The condition becomes (a⃗ + 2b⃗) × c⃗ + 3a⃗ × c⃗ = 0⃗.

   (4a⃗ + 2b⃗) × c⃗ = 0⃗ ⇒ (2a⃗ + b⃗) × c⃗ = 0⃗

2. Parallel vectors. A zero cross product means c⃗ is parallel to 2a⃗ + b⃗, so c⃗ = λ(2a⃗ + b⃗), where 2a⃗ + b⃗ = 4î - 7ĵ + 15k̂.
3. Find lambda. a⃗ · (2a⃗ + b⃗) = 4 + 21 + 105 = 130. Since a⃗ · c⃗ = 130, we get 130λ = 130, so λ = 1 and c⃗ = 4î - 7ĵ + 15k̂.
4. So. b⃗ · c⃗ = (2)(4) + (-1)(-7) + (1)(15) = 8 + 7 + 15 = 30.

The idea to keep. When a cross-product equation has the same vector c⃗ on both sides, bring all terms to one side. u⃗ × c⃗ = 0⃗ gives c⃗ = λu⃗, and the dot-product condition fixes λ.

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