Let a⃗ and b⃗ be two vectors such that |b⃗|=1 and |b⃗ × a⃗|=2. Then |(b⃗ × a⃗)-b⃗|² is equal to

Solution

Maths sir · 4 steps

A cross product is perpendicular to its own vectors, so the middle term of the square is zero.

1. Name the vector. Let c⃗ = b⃗ × a⃗. We are told |c⃗| = 2 and |b⃗| = 1, and we need |c⃗ - b⃗|².

   |c⃗ - b⃗|² = |c⃗|² + |b⃗|² - 2 c⃗ · b⃗

2. Middle term. A cross product is perpendicular to both vectors in it. So c⃗ = b⃗ × a⃗ is perpendicular to b⃗, which gives c⃗ · b⃗ = 0.
3. Put in numbers. |c⃗ - b⃗|² = 2² + 1² - 2 × 0 = 4 + 1 = 5.
4. So. |(b⃗ × a⃗) - b⃗|² = 5, which is option (c).

The idea to keep. Whenever a cross product is dotted with one of its own vectors, the answer is zero. Expand |p⃗ - q⃗|² and look for that perpendicular pair first.

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