Let a⃗ and b⃗ be two vectors such that |a⃗|=1,|b⃗|=4, and a⃗ · b⃗=2. If c⃗=(2 a⃗ × b⃗)-3 b⃗ and the angle between b⃗ and c⃗ is α, then 192 sin ² α is equal to ________.

Solution

Maths sir · 4 steps

The cross product is perpendicular to b⃗, so most terms vanish when you find cos α.

1. Dot c with b. a⃗×b⃗ is perpendicular to b⃗, so (a⃗×b⃗)·b⃗ = 0. That leaves b⃗·c⃗ = -3|b⃗|² = -3 × 16 = -48.
2. Length of c. The two parts of c⃗ are perpendicular, so their squared lengths add. First, |a⃗×b⃗|² = |a⃗|²|b⃗|² - (a⃗·b⃗)² = 16 - 4 = 12.

   |c⃗|² = 4|a⃗×b⃗|² + 9|b⃗|² = 4 × 12 + 9 × 16 = 192

3. Find the angle. cos² α = (b⃗·c⃗)²/|b⃗|²|c⃗|² = 2304/(16 × 192) = 3/4. So sin² α = 1 - 3/4 = 1/4.
4. So. 192 sin² α = 192 × 1/4 = 48.

The common slip: 144. Stopping at cos² α = 3/4 and multiplying that by 192 gives 144. The question asks for sin² α, which is 1/4.

The idea to keep. When a vector is built from a⃗×b⃗ and b⃗, use the fact that they are perpendicular. Cross terms drop out, and |a⃗×b⃗|² = a²b² - (a⃗·b⃗)² gives the rest.

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