Let the unit vectors a and b be perpendicular to each other and the unit vector c be inclined at an angle θ to both a and b. If c = x a + y b + z (a × b), then what are the values of x, y, and z?

We have c = xa + yb + z(a × b) ⇒ c.a = x and c.b = y ⇒ x = y = cosθ
Now, c.c = |c|² ⇒ (xa + yb + z(a × b)) (xa + yb + z(a × b)) = |c|²
⇒ x² + y² + z²|a × b|² = 1 since a and b are unit vectors.
⇒ x² + y² + z² = 1 since |a × b| = 1 (a and b are perpendicular unit vectors)
⇒ 2x² + z² = 1 (since x = y = cosθ)
⇒ z² = 1 - 2x² = 1 - 2cos²θ = 1 - (1 + cos2θ) = -cos2θ