devarshi-dt-logo

Question:

Let 𝐚 and 𝐛 be two unit vectors such that 𝐚⋅𝐛 = 0. For some x, y ∈ R, let 𝐜 = x𝐚 + y𝐛 + (𝐚 × 𝐛). If |𝐜| = 2 and the vector 𝐜 is inclined at the same angle α to both 𝐚 and 𝐛, then the value of 8cos²α is __________.

Solution:

𝐜 = x𝐚 + y𝐛 + 𝐚 × 𝐛
𝐜 ⋅ 𝐚 = x and x = 2cosα
𝐜 ⋅ 𝐛 = y and y = 2cosα
Also, |𝐚 × 𝐛| = 1
∴ 𝐜 = (2cosα)(𝐚 + 𝐛) + 𝐚 × 𝐛
∴ |𝐜|² = 4cos²α(𝐚 + 𝐛)² + (𝐚 × 𝐛)² + 2cosα(𝐚 + 𝐛) ⋅ (𝐚 × 𝐛)
∴ 4 = 8cos²α + 1
∴ 8cos²α = 3