Let a and b be two unit vectors such that the angle between them is π/4. If θ is the angle between the vectors (a+b) and (a+2 b+2(a × b)), then the value of 164 cos ² θ is equal to :

Solution

Maths sir · 5 steps

The cross product part is perpendicular to a and b, so it drops out of the dot product.

1. Basic values. With the angle π/4, a·b = 1/√2 and |a×b|² = sin² π/4 = 1/2. Also a×b is perpendicular to both a and b, so every dot product with it is 0.
2. Dot product. Let u⃗ = a+b and v⃗ = a+2b+2(a×b). Then u⃗·v⃗ = 1 + 2a·b + a·b + 2 = 3 + 3/√2, so (u⃗·v⃗)² = 9(3/2+√2) = (27+18√2)/2.
3. Magnitudes. |u⃗|² = 1+1+√2 = 2+√2. Since a+2b is perpendicular to the cross product part, |v⃗|² = (1+4+2√2) + 4×1/2 = 7+2√2.

   cos² θ = (27+18√2)/2(2+√2)(7+2√2) = (27+18√2)/2(18+11√2)

4. Rationalise. Multiply top and bottom by 18-11√2. The denominator becomes 2(324-242) = 164, and the numerator becomes 486 - 297√2 + 324√2 - 396 = 90+27√2.
5. So. cos² θ = (90+27√2)/164, so 164 cos² θ = 90+27√2, which is option (a).

The idea to keep. When a vector contains a×b, use that it is perpendicular to a and b. It drops out of dot products, and its square adds separately to the magnitude.

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