Let θ be the angle between the vectors a⃗ and b⃗, where |a⃗ | = 4, |b⃗ | = 3 and θ ∈ ( π/4,π/3 ). Then | ( a⃗ - b⃗ ) × ( a⃗ + b⃗ ) |² + 4( a⃗ . b⃗ )² is equal to __________.

Solution

Maths sir · 4 steps

Expand the cross product first: it becomes 2 a⃗ × b⃗, and then the angle disappears.

1. Expand the cross product. (a⃗ - b⃗) × (a⃗ + b⃗) = a⃗ × a⃗ + a⃗ × b⃗ - b⃗ × a⃗ - b⃗ × b⃗. Since a⃗ × a⃗ = b⃗ × b⃗ = 0⃗ and b⃗ × a⃗ = -a⃗ × b⃗, this is 2 a⃗ × b⃗.
2. Square the magnitude. So the first term is |2 a⃗ × b⃗|² = 4|a⃗ × b⃗|². The whole expression is 4(|a⃗ × b⃗|² + (a⃗ · b⃗)²).

   |a⃗ × b⃗|² + (a⃗ · b⃗)² = |a⃗|²|b⃗|²(sin² θ + cos² θ) = |a⃗|²|b⃗|²

3. Put in numbers. |a⃗|²|b⃗|² = 16 × 9 = 144. The value of θ does not matter, so the given range of θ plays no part.
4. So. The expression equals 4 × 144 = 576.

The idea to keep. Whenever |a⃗ × b⃗|² and (a⃗ · b⃗)² appear together with equal weight, add them: sin² θ + cos² θ = 1 leaves just |a⃗|²|b⃗|².

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