Let a⃗ = î + 2ĵ - 3k̂ and b⃗ = 2î - 3ĵ + 5k̂. If r⃗ × a⃗ = b⃗ × r⃗,
r⃗ . ( α î + 2ĵ + k̂ ) = 3 and r⃗ . ( 2î + 5ĵ - α k̂ ) = -1, α ∈ R, then the
value of α + | r⃗ |² is equal to :

Solution

Maths sir · 4 steps

Flip b⃗ × r⃗ to -r⃗ × b⃗, and r⃗ must be parallel to a⃗ + b⃗.

1. Bring to one side. Since b⃗ × r⃗ = -r⃗ × b⃗, the condition becomes r⃗ × a⃗ + r⃗ × b⃗ = 0⃗, that is r⃗ × (a⃗ + b⃗) = 0⃗. A zero cross product means r⃗ is parallel to a⃗ + b⃗ = 3î - ĵ + 2k̂.

   r⃗ = λ(3î - ĵ + 2k̂)

2. Use both dot products. First: λ(3α - 2 + 2) = 3, so λα = 1. Second: λ(6 - 5 - 2α) = -1, so λ - 2λα = -1, which gives λ - 2 = -1, so λ = 1 and α = 1.
3. Find the length. With λ = 1, r⃗ = 3î - ĵ + 2k̂, so |r⃗|² = 9 + 1 + 4 = 14.
4. So. α + |r⃗|² = 1 + 14 = 15, which is option (d).

The idea to keep. Whenever r⃗ × u⃗ = r⃗ × v⃗ or a swapped version appears, rearrange to r⃗ × (·) = 0⃗. Then r⃗ is a scalar multiple, and the dot conditions fix the scalar.

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