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JEE Main 2017 · Mathematics · Vector Algebra · medium

If vec a=hat i+2hat j+3hat k and vec b=2hat i-hat j+hat k, then |vec a×vec b| equals

Single correct option

If a⃗=i^+2j^+3k^\vec a=\hat i+2\hat j+3\hat k and b⃗=2i^−j^+k^\vec b=2\hat i-\hat j+\hat k, then ∣a⃗×b⃗∣|\vec a\times\vec b| equals
  1. A. 535\sqrt3
  2. B. 353\sqrt5
  3. C. 75+3\sqrt{75}+3
  4. D. 1515
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Answer: (A) 535\sqrt3

Expand the determinant: a⃗×b⃗=(2⋅1−3⋅(−1))i^−(1⋅1−3⋅2)j^+(1⋅(−1)−2⋅2)k^=5i^+5j^−5k^\vec a\times\vec b=(2\cdot1-3\cdot(-1))\hat i-(1\cdot1-3\cdot2)\hat j+(1\cdot(-1)-2\cdot2)\hat k=5\hat i+5\hat j-5\hat k. Its magnitude is 25+25+25=75=53\sqrt{25+25+25}=\sqrt{75}=5\sqrt3. As a check, a⃗⋅b⃗=3\vec a\cdot\vec b=3 and ∣a⃗∣2∣b⃗∣2−(a⃗⋅b⃗)2=84−9=75|\vec a|^2|\vec b|^2-(\vec a\cdot\vec b)^2=84-9=75.
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