devarshi-dt-logo

Question:

Let 𝐚, 𝐛, and 𝐜 be three non-zero vectors such that no two of them are collinear and (𝐚 × 𝐛) × 𝐜 = 13|𝐛||𝐜|𝐚. If θ is the angle between vectors 𝐛 and 𝐜, then a value of sin θ is

−√23

√33

2√23

23

Solution:

(𝐚 × 𝐛) × 𝐜 = (𝐚 ⋅ 𝐜)𝐛 − (𝐛 ⋅ 𝐜)𝐚 = 13|𝐛||𝐜|𝐚
By comparing, 𝐛 ⋅ 𝐜 = −13|𝐛||𝐜| ⇒ cos θ = −13 ⇒ sin θ = √1 − cos²θ = √1 − (13)² = √1 − 19 = √89 = 2√23