Let a⃗=2 î+3 ĵ+4 k̂, b⃗=î-2 ĵ-2 k̂ and c⃗=-î+4 ĵ+3 k̂. If d⃗ is a vector perpendicular to both b⃗ and c⃗, and a⃗ · d⃗=18, then |a⃗ × d⃗|² is equal to :

Solution

Maths sir · 4 steps

Find d⃗ from b⃗ × c⃗, then use Lagrange's identity instead of a second cross product.

1. Direction of d. A vector perpendicular to both b⃗ and c⃗ is parallel to b⃗ × c⃗. Expanding the determinant gives b⃗ × c⃗ = (6 · (-1) + 8)î - (3 - 2)ĵ + (4 - 2)k̂ = 2î - ĵ + 2k̂.
2. Fix the scale. Take d⃗ = λ(2î - ĵ + 2k̂). Then a⃗ · d⃗ = λ(4 - 3 + 8) = 9λ = 18, so λ = 2 and d⃗ = 4î - 2ĵ + 4k̂, with |d⃗|² = 36.

   |a⃗ × d⃗|² = |a⃗|² |d⃗|² - (a⃗ · d⃗)²

3. Put in numbers. |a⃗|² = 4 + 9 + 16 = 29. So |a⃗ × d⃗|² = 29 × 36 - 18² = 1044 - 324 = 720.
4. So. |a⃗ × d⃗|² = 720, which is option (b).

The idea to keep. Perpendicular to two vectors means parallel to their cross product. When you know a⃗ · d⃗, get |a⃗ × d⃗|² from |a⃗|²|d⃗|² - (a⃗ · d⃗)².

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