Let a⃗ = 2î + ĵ - 2k̂ and b⃗ = î + ĵ. If c⃗ is a vector such that a⃗ . c⃗ = | c⃗ |,| c⃗ - a⃗ | = 2√2 and the angle between (a⃗ × b⃗ ) and c⃗ is π/6, then the value of | ( a⃗ × b⃗ ) × c⃗ | is :

Solution

Maths sir · 3 steps

Square |c⃗ - a⃗| to find |c⃗|, then use |u⃗||c⃗|sin θ.

1. Length of c. Here |a⃗| = √(4+1+4) = 3. Square the second condition to get |c⃗|² - 2 a⃗·c⃗ + |a⃗|² = 8. Now put a⃗·c⃗ = |c⃗| and |a⃗|² = 9.

   |c⃗|² - 2|c⃗| + 1 = 0 ⇒ (|c⃗| - 1)² = 0 ⇒ |c⃗| = 1

2. Find a cross b. a⃗ × b⃗ = |î, ĵ, k̂; 2, 1, -2; 1, 1, 0| = 2î - 2ĵ + k̂. Its magnitude is √(4+4+1) = 3.
3. So. |(a⃗ × b⃗)× c⃗| = 3 × 1 × sin π/6 = 3 × 1/2 = 3/2. That is option (d).

The idea to keep. When a condition gives |c⃗ - a⃗|, square it and expand. It usually turns into a quadratic in |c⃗|. The magnitude of any cross product is |u⃗||v⃗|sin θ.

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