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

Solution

Maths sir · 4 steps

Expand the cross product first: everything collapses to a multiple of a⃗ × b⃗.

1. Expand. Multiply term by term. The terms a⃗ × a⃗ and b⃗ × b⃗ are zero, so only a⃗ × (-3b⃗) and 2b⃗ × 2a⃗ survive.

   (a⃗+2b⃗) × (2a⃗-3b⃗) = -3 a⃗ × b⃗ + 4 b⃗ × a⃗ = -7 a⃗ × b⃗

2. Flip the order. Use b⃗ × a⃗ = -a⃗ × b⃗. That turns +4 into -4, so the total is -3 - 4 = -7.
3. Find the magnitude. |a⃗ × b⃗|² = |a⃗|² |b⃗|² sin² π/4 = 4 × 9 × 1/2 = 18.
4. So. The required value is (-7)² × 18 = 49 × 18 = 882, option (d).

The idea to keep. A cross product of two combinations of a⃗ and b⃗ is always a number times a⃗ × b⃗. Watch the sign when you flip b⃗ × a⃗.

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