Let a⃗ = α î + 3ĵ - k̂, b⃗ = 3î - β ĵ + 4k̂ and c⃗ = î + 2ĵ - 2k̂ where α , β ∈ R, be three vectors. If the projection of a⃗ on c⃗ is 10/3 and b⃗ × c⃗ = - 6î + 10ĵ + 7k̂, then the value of α + β is equal to :

Solution

Maths sir · 4 steps

Use each condition to pin one unknown: the projection gives α, and the cross product gives β.

1. Projection gives alpha. The projection of a⃗ on c⃗ is (a⃗ · c⃗)/|c⃗|. Here a⃗ · c⃗ = α + 6 + 2 = α + 8 and |c⃗| = √(1 + 4 + 4) = 3.

   (α + 8)/3 = 10/3 ⇒ α = 2

2. Cross product gives beta. Expand b⃗ × c⃗ with b⃗ = (3, -β, 4) and c⃗ = (1, 2, -2). The components are (2β - 8)î + 10ĵ + (6 + β)k̂.
3. Match components. Compare with -6î + 10ĵ + 7k̂. The î part gives 2β - 8 = -6, so β = 1, and the k̂ part gives 6 + β = 7, which agrees.
4. So. α + β = 2 + 1 = 3, which is option (a).

The idea to keep. Projection of a⃗ on c⃗ means divide the dot product by |c⃗|, not |c⃗|². When a cross product is given, match components and use a spare one to check your answer.

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