Let a⃗=2 î+5 ĵ-k̂, b⃗=2 î-2 ĵ+2 k̂ and c⃗ be three vectors such that (c⃗+î) ×(a⃗+b⃗+î)=a⃗ ×(c⃗+î). If a⃗ · c⃗=-29, then c⃗ ·(-2 î+ĵ+k̂) 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 right side. Swapping the order of a cross product flips its sign, so a⃗ × (c⃗+î) = -(c⃗+î) × a⃗. Bring it to the left side and take (c⃗+î) out as a common factor.

   (c⃗+î) × (2a⃗+b⃗+î) = 0⃗

2. Parallel vectors. Here 2a⃗+b⃗+î = (4+2+1)î + (10-2)ĵ + (-2+2)k̂ = 7î+8ĵ. A zero cross product means c⃗+î = λ(7î+8ĵ), so c⃗ = (7λ-1)î + 8λĵ.
3. Use the dot product. a⃗ · c⃗ = 2(7λ-1) + 5(8λ) = 54λ - 2. Setting this equal to -29 gives λ = -1/2, so c⃗ = -9/2î - 4ĵ.
4. So. c⃗ · (-2î+ĵ+k̂) = 9 - 4 + 0 = 5, which is option (c).

The idea to keep. When the same vector appears in cross products on both sides, collect everything into one cross product. If that equals zero, the two vectors are parallel, and one unknown λ replaces three.

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