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JEE Advanced 2022 Paper 1 · Mathematics · Vector Algebra · medium

Let vec a=hat i+2hat j-hat k, vec b=2hat i-hat j+hat k, vec c=hat i+hat j+3hat k.

One or more options correct

Let a⃗=i^+2j^−k^, b⃗=2i^−j^+k^, c⃗=i^+j^+3k^\vec a=\hat i+2\hat j-\hat k,\ \vec b=2\hat i-\hat j+\hat k,\ \vec c=\hat i+\hat j+3\hat k. Which of the following statements is/are correct?
  1. A. a⃗\vec a is perpendicular to c⃗\vec c.
  2. B. The volume of the parallelepiped with edges a⃗,b⃗,c⃗\vec a,\vec b,\vec c is 1717.
  3. C. ∣a⃗×b⃗∣=34|\vec a\times\vec b|=\sqrt{34}.
  4. D. a⃗×(b⃗×c⃗)=3b⃗\vec a\times(\vec b\times\vec c)=3\vec b.
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Answer: A, B

(A) a⃗⋅c⃗=1+2−3=0\vec a\cdot\vec c=1+2-3=0: true. (B) a⃗⋅(b⃗×c⃗)=∣12−12−11113∣=1(−4)−2(5)+(−1)(3)=−17\vec a\cdot(\vec b\times\vec c)=\begin{vmatrix}1&2&-1\\2&-1&1\\1&1&3\end{vmatrix}=1(-4)-2(5)+(-1)(3)=-17, so the volume is 1717: true. (C) a⃗×b⃗=(1,−3,−5)\vec a\times\vec b=(1,-3,-5) so ∣a⃗×b⃗∣=35≠34|\vec a\times\vec b|=\sqrt{35}\ne\sqrt{34}: false. (D) a⃗×(b⃗×c⃗)=(a⃗⋅c⃗)b⃗−(a⃗⋅b⃗)c⃗=0⋅b⃗+c⃗=c⃗=(1,1,3)\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c=0\cdot\vec b+\vec c=\vec c=(1,1,3) since a⃗⋅b⃗=−1\vec a\cdot\vec b=-1; this is not 3b⃗3\vec b: false.
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Original question written in the JEE Advanced 2022 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake