Let a, b be unit vectors. If c⃗ be a vector such that the angle between a and c⃗ is π/12, and b = c⃗ + 2( c⃗ × a ), then | 6c⃗ |² is equal to :

Solution

Maths sir · 4 steps

Square the modulus of both sides: c⃗ and c⃗ × a are perpendicular.

1. Square both sides. Take |·|² of b = c⃗ + 2(c⃗ × a). The vector c⃗ × a is perpendicular to c⃗, so c⃗ · (c⃗ × a) = 0 and the cross term disappears.

   1 = |b|² = |c⃗|² + 4|c⃗ × a|² = |c⃗|²(1 + 4 sin² π/12)

2. Use the angle. Here |c⃗ × a| = |c⃗|sin π/12, since |a| = 1. Also sin² 15° = (1 - cos 30°)/2 = (2 - √3)/4, so 4 sin² 15° = 2 - √3.
3. Find c squared. 1 = |c⃗|² (3 - √3), so |c⃗|² = 1/(3 - √3) = (3 + √3)/6 after rationalising.
4. So. |6c⃗|² = 36|c⃗|² = 36 × (3 + √3)/6 = 6(3 + √3), which is option (c).

The idea to keep. When a vector equation mixes c⃗ and c⃗ × a, square the modulus of both sides. The two parts are perpendicular, so the lengths add like Pythagoras.

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