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NDA 2021 Maths · Mathematics · Vector Algebra · hard

If |vec a|=3, |vec b|=4, then |vec a×vec b|^2+(vec a·vec b)^2 is equal to

Single correct option

If ∣a⃗∣=3|\vec a|=3, ∣b⃗∣=4|\vec b|=4, then ∣a⃗×b⃗∣2+(a⃗⋅b⃗)2|\vec a\times\vec b|^2+(\vec a\cdot\vec b)^2 is equal to
  1. A. 144144
  2. B. 1212
  3. C. 2525
  4. D. 4949
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Answer: (A) 144144

Since ∣a⃗×b⃗∣2=∣a⃗∣2∣b⃗∣2sin⁡2θ|\vec a\times\vec b|^2=|\vec a|^2|\vec b|^2\sin^2\theta and (a⃗⋅b⃗)2=∣a⃗∣2∣b⃗∣2cos⁡2θ(\vec a\cdot\vec b)^2=|\vec a|^2|\vec b|^2\cos^2\theta, the sum is ∣a⃗∣2∣b⃗∣2=9⋅16=144|\vec a|^2|\vec b|^2=9\cdot16=144 whatever the angle. The value 12 is ∣a⃗∣∣b⃗∣|\vec a||\vec b| before squaring.
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