Let a⃗ = α î + 2ĵ - k̂ and b⃗ = - 2î + α ĵ + k̂, where α ∈ R. If the area of the parallelogram whose adjacent sides are represented by the vectors a⃗ and b⃗ is √15(α ² + 4), then the value of 2| a⃗ |² + ( a⃗ . b⃗ )| b⃗ |² is equal to :

Solution

Maths sir · 4 steps

Square the area, factor out α² + 4, and α² drops out in one line.

1. Cross product. With a⃗ = (α, 2, -1) and b⃗ = (-2, α, 1), the cross product is a⃗ × b⃗ = (2 + α) î + (2 - α) ĵ + (α² + 4) k̂.
2. Square the area. The area is |a⃗ × b⃗|. Since (2+α)² + (2-α)² = 2(α² + 4), every term carries the factor α² + 4.

   2(α²+4) + (α²+4)² = 15(α²+4) ⇒ 2 + (α²+4) = 15 ⇒ α² = 9

3. Find each piece. |a⃗|² = α² + 4 + 1 = 14 and |b⃗|² = 4 + α² + 1 = 14. Also a⃗ · b⃗ = -2α + 2α - 1 = -1, whatever the sign of α.
4. So. 2|a⃗|² + (a⃗ · b⃗)|b⃗|² = 2 × 14 + (-1) × 14 = 14, which is option (d).

The idea to keep. Compare squared areas, not square roots. Look for a common factor such as α² + 4 to divide out. You only need α², never α itself.

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