Let a⃗, b⃗ and c⃗ be three vectors such that | a⃗ | = √3, | b⃗ | = 5,b⃗ .c⃗ = 10 and the angle between b⃗ and c⃗ is π/3. If a⃗ is perpendicular to the vector b⃗ × c⃗ , then | a⃗ × ( b⃗ × c⃗ ) | is equal to _____.

Solution

Maths sir · 4 steps

Treat b⃗ × c⃗ as one vector; a⃗ is perpendicular to it, so the sine is just 1.

1. Length of c. b⃗ · c⃗ = |b⃗||c⃗|cos π/3, so 10 = 5 × |c⃗| × 1/2. That gives |c⃗| = 4.
2. Cross product length. |b⃗ × c⃗| = |b⃗||c⃗|sin π/3 = 5 × 4 × √3/2 = 10√3.
3. Use the right angle. Call d⃗ = b⃗ × c⃗. The angle between a⃗ and d⃗ is 90°, so sin 90° = 1. The magnitude of the cross product is then the product of the two lengths.

   |a⃗ × (b⃗ × c⃗)| = |a⃗| |b⃗ × c⃗|sin 90° = √3 × 10√3

4. So. √3 × 10√3 = 10 × 3, so the answer is 30.

The common slip: 0. Reading "perpendicular" as making the cross product vanish gives 0. Perpendicular makes the dot product zero and makes the cross product as large as possible.

The idea to keep. You do not always need to expand a triple product. If you know the angle between the outer vector and the inner cross product, use |p⃗ × q⃗| = |p⃗||q⃗|sin θ directly.

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