Let a⃗=α î+ĵ+β k̂ and b⃗=3 î-5 ĵ+4 k̂ be two vectors, such that a⃗ × b⃗=-î+9 ĵ+12 k̂. Then the projection of b⃗-2 a⃗ on b⃗+a⃗ is equal to :

Solution

Maths sir · 4 steps

Match the cross product part by part to find α and β, then divide one dot product by one length.

1. Expand the cross product. With a⃗=(α, 1, β) and b⃗=(3, -5, 4), the determinant gives a⃗×b⃗ = (4+5β) î + (3β-4α) ĵ + (-5α-3) k̂.
2. Find alpha and beta. Match with -î+9ĵ+12k̂. From 4+5β=-1, β=-1. From -5α-3=12, α=-3, and the check 3β-4α=-3+12=9 holds, so a⃗=-3î+ĵ-k̂.
3. Build both vectors. b⃗-2a⃗ = 9î-7ĵ+6k̂ and b⃗+a⃗ = 0î-4ĵ+3k̂, whose length is √(16+9)=5.

   Projection = ((b⃗-2a⃗)·(b⃗+a⃗))/(|b⃗+a⃗|) = (0+28+18)/5

4. So. The projection is 46/5, which is option (d).

The idea to keep. Equate the components of a given cross product to find the unknowns, and use the leftover component as a check. The projection of p⃗ on q⃗ is (p⃗·q⃗)/|q⃗|.

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