If | a⃗ | = 2,| b⃗ | = 5 and | a⃗ × b⃗ | = 8, then | a⃗ . b⃗ | is equal to :

Solution

Maths sir · 3 steps

Square both products and add them: the angle drops out and leaves |a⃗|²|b⃗|².

1. Write both in θ. |a⃗ × b⃗| = |a⃗||b⃗|sin θ and |a⃗ · b⃗| = |a⃗||b⃗||cos θ|. Squaring and adding uses sin² θ + cos² θ = 1.

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

2. Put in numbers. 8² + |a⃗ · b⃗|² = 2² × 5², so 64 + |a⃗ · b⃗|² = 100 and |a⃗ · b⃗|² = 36.
3. So. |a⃗ · b⃗| = 6, which is option (a).

The idea to keep. Whenever you know one of |a⃗ × b⃗| or a⃗ · b⃗ and the two lengths, use Lagrange's identity: their squares add up to |a⃗|²|b⃗|².

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