Let a⃗=3 î+ĵ-2 k̂, b⃗=4 î+ĵ+7 k̂ and c⃗=î-3 ĵ+4 k̂ be three vectors. If a vectors p⃗ satisfies p⃗ × b⃗=c⃗ × b⃗ and p⃗ · a⃗=0, then p⃗ ·(î-ĵ-k̂) is equal to

Solution

Maths sir · 4 steps

Same cross product with b⃗ means p⃗-c⃗ is parallel to b⃗. Start there.

1. Use the cross product. Move everything to one side: (p⃗ - c⃗) × b⃗ = 0⃗. A zero cross product means p⃗ - c⃗ is parallel to b⃗, so p⃗ = c⃗ + λb⃗ for some scalar λ.
2. Use the dot product. Dot with a⃗ and set it to zero: c⃗·a⃗ + λ b⃗·a⃗ = 0. Here c⃗·a⃗ = 3 - 3 - 8 = -8 and b⃗·a⃗ = 12 + 1 - 14 = -1.

   -8 - λ = 0 ⇒ λ = -8

3. Find p. p⃗ = c⃗ - 8b⃗ = (1 - 32)î + (-3 - 8)ĵ + (4 - 56)k̂ = -31î - 11ĵ - 52k̂.
4. So. p⃗·(î - ĵ - k̂) = -31 + 11 + 52 = 32, which is option (b).

The idea to keep. Whenever x⃗ × b⃗ = y⃗ × b⃗, write x⃗ = y⃗ + λb⃗. The extra dot-product condition then fixes λ.

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