MCQ
If $\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}},\vec{\text{b}}=3\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ and $\vec{\text{c}}=5\hat{\text{i}}-3\hat{\text{j}}-2\hat{\text{k}},$ then the volume of the parallelopiped with contermious edges $\vec{\text{a}}+\vec{\text{b}},\vec{\text{b}}+\vec{\text{c}},\vec{\text{c}}+\vec{\text{a}}$ is:
  • A
    $2$
  • B
    $1$
  • C
    $-1$
  • None  of these

Answer

Correct option: D.
None  of these
We have
$\vec{\text{a}}+\vec{\text{b}}=\big(2\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}}\big)+\big(3\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}\big)=5\hat{\text{i}}-7\hat{\text{j}}+10\hat{\text{k}}$
$\vec{\text{b}}+\vec{\text{c}}=\big(3\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}\big)\big(5\hat{\text{i}}-3\hat{\text{j}}-2\hat{\text{k}}\big)=8\hat{\text{i}}-7\hat{\text{j}}+3\hat{\text{k}}$
$\vec{\text{c}}+\vec{\text{a}}=\big(5\hat{\text{i}}-3\hat{\text{j}}-2\hat{\text{k}}\big)+\big(2\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}}\big)=7\hat{\text{i}}-6\hat{\text{j}}+3\hat{\text{k}}$
We know that the volume of parallelopiped whose three adjacent adges are $\vec{\text{a}}+\vec{\text{b}},\vec{\text{b}}+\vec{\text{c}}$ and $\vec{\text{c}}+\vec{\text{a}}$ is equal to:
We have
$\big[\vec{\text{a}}+\vec{\text{b}}\vec{\text{b}}+\vec{\text{c}}\vec{\text{c}}+\vec{\text{a}}\big]=\begin{vmatrix}5&-7&10\\8&-7&3\\7&-6&3\end{vmatrix}$
$=5(-21+18)+7(24-21)+10(-48+49)$
$=(5\times-3)+(7\times3)+(10\times1)$
$=16$
$\therefore$ volume of parallelopiped $=\big|16\big|=16$
Disclaimer: None of the given option is correct.

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