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Question:

If 𝐚, 𝐛, 𝐜 and 𝐝 are unit vectors such that (𝐚 × 𝐛) ⋅ (𝐜 × 𝐝) = 1 and 𝐚 ⋅ 𝐜 = 1/2, then

𝐚, 𝐛, 𝐜 and 𝐝 are coplanar vectors

𝐚, 𝐝 are parallel and 𝐛, 𝐜 are parallel

𝐚, 𝐛, 𝐜 are non-coplanar

𝐛, 𝐜, 𝐝 are non-coplanar

Solution:

(𝐚 × 𝐛) ⋅ (𝐜 × 𝐝) = |𝐚 × 𝐛| |𝐜 × 𝐝| cos θ = 1
Hence
For this to be true, (𝐚 × 𝐛) || (𝐜 × 𝐝) and |𝐚 × 𝐛| = |𝐜 × 𝐝| = 1
Hence 𝐚, 𝐛, 𝐜, 𝐝 are coplanar vectors.