Let a vector a⃗ has magnitude 9. Let a vector b⃗ be such that for every (x, y) ∈ R × R-{(0,0)}, the vector (x a⃗+y b⃗) is perpendicular to the vector (6 y a⃗-18 x b⃗). Then the value of |a⃗ × b⃗| is equal to :

Solution

Maths sir · 4 steps

True for every x and y means each coefficient of the expanded dot product must vanish on its own.

1. Set dot product zero. Perpendicular vectors have zero dot product. Expand (xa⃗+yb⃗)·(6ya⃗-18xb⃗) term by term, using a⃗·a⃗=|a⃗|² and a⃗·b⃗=b⃗·a⃗.

   -18 (a⃗·b⃗) x² + (6|a⃗|²-18|b⃗|²)xy + 6 (a⃗·b⃗) y² = 0

2. Every coefficient zero. This holds for all (x, y), so each coefficient is zero. The x² and y² terms give a⃗·b⃗=0, so a⃗⊥b⃗. The xy term gives 6|a⃗|²=18|b⃗|², so |b⃗|²=81/3=27 and |b⃗|=3√3.
3. Cross product size. Since a⃗⊥b⃗, sin θ=1. So |a⃗×b⃗|=|a⃗||b⃗|=9× 3√3.
4. So. |a⃗×b⃗|=27√3, which is option (b).

The idea to keep. When an expression in x and y is zero for every x and y, compare coefficients: each coefficient must be zero separately. That turns one condition into several equations.

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