Let a⃗ and b⃗ be two vectors such that |a⃗|=√14,|b⃗|=√6 and |a⃗ × b⃗|=√48. Then (a⃗ · b⃗)² is equal to ___________.

Solution

Maths sir · 3 steps

Dot and cross product squared always add up to |a⃗|²|b⃗|². Use that and you are done.

1. Use the identity. With θ the angle between the vectors, a⃗ · b⃗ = |a⃗||b⃗|cos θ and |a⃗ × b⃗| = |a⃗||b⃗|sin θ. Square both and add. Since sin² θ + cos² θ = 1, you get Lagrange's identity.

   |a⃗ × b⃗|² + (a⃗ · b⃗)² = |a⃗|²|b⃗|²

2. Put in numbers. |a⃗|²|b⃗|² = 14 × 6 = 84 and |a⃗ × b⃗|² = 48. So 48 + (a⃗ · b⃗)² = 84.
3. So. (a⃗ · b⃗)² = 84 - 48 = 36.

The common slip: 132. Adding |a⃗ × b⃗|² to |a⃗|²|b⃗|² instead of subtracting it gives 132, not 36.

The idea to keep. Whenever you know two of |a⃗||b⃗|, a⃗ · b⃗ and |a⃗ × b⃗|, square them. The squares of the dot and cross products add up to |a⃗|²|b⃗|².

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