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JEE Main 2021 · Mathematics · Vector Algebra · medium

If vec a,vec b are vectors with |vec a|=2, |vec b|=3 and vec a·vec b=3, then |vec a×vec b| equals

Single correct option

If a⃗,b⃗\vec a,\vec b are vectors with ∣a⃗∣=2|\vec a|=2, ∣b⃗∣=3|\vec b|=3 and a⃗⋅b⃗=3\vec a\cdot\vec b=3, then ∣a⃗×b⃗∣|\vec a\times\vec b| equals
  1. A. 33
  2. B. 99
  3. C. 353\sqrt5
  4. D. 333\sqrt3
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Answer: (D) 333\sqrt3

Use Lagrange's identity ∣a⃗×b⃗∣2=∣a⃗∣2∣b⃗∣2−(a⃗⋅b⃗)2=4⋅9−9=27|\vec a\times\vec b|^2=|\vec a|^2|\vec b|^2-(\vec a\cdot\vec b)^2=4\cdot9-9=27. Therefore ∣a⃗×b⃗∣=33|\vec a\times\vec b|=3\sqrt3. Equivalently cos⁡θ=12\cos\theta=\tfrac12, so sin⁡θ=32\sin\theta=\tfrac{\sqrt3}{2} and ∣a⃗×b⃗∣=6⋅32|\vec a\times\vec b|=6\cdot\tfrac{\sqrt3}{2}.
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