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Question Id : 11625 | Context :NIMCET 2024

Question

The value of m for which volume of the parallelepiped is 4 cubic units whose three edges are represented by a = mi + j + k, b = i – j + k, c = i + 2j –k is
🎥 Video solution / Text Solution of this question is given below:

Given: Volume of a parallelepiped formed by vectors →a,→b,→c is 4 cubic units.

Vectors:

  • →a=mˆi+ˆj+ˆk
  • →b=ˆi−ˆj+ˆk
  • →c=ˆi+2ˆj−ˆk

Step 1: Volume = |→a⋅(→b×→c)|

First compute →b×→c:

→b×→c=|ˆiˆjˆk1−1112−1|=ˆi((−1)(−1)−(1)(2))−ˆj((1)(−1)−(1)(1))+ˆk((1)(2)−(−1)(1))=ˆi(1−2)−ˆj(−1−1)+ˆk(2+1)=−ˆi+2ˆj+3ˆk

Step 2: Compute dot product with →a:

→a⋅(→b×→c)=(m)(−1)+(1)(2)+(1)(3)=−m+2+3=−m+5

Step 3: Volume = |−m+5|=4

So, |−m+5|=4⇒−m+5=±4

  • Case 1: −m+5=4⇒m=1
  • Case 2: −m+5=−4⇒m=9

✅ Final Answer: m=1 or 9

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