If the volume of parallelopiped with conterminus edges $4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}},-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}+p \hat{\mathbf{k}}$ is 34 cubic units, then $p$ is equal to :

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Aicharya · Accepted Answer

Coterminus edges of a parallelopiped are $4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}},-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}+p \hat{\mathbf{k}}$ Volume of parallelopiped $=34$ $\Rightarrow \quad\left|\begin{array}{rrr}4 & 5 & 1 \ 0 & -1 & 1 \ 3 & 9 & p\end{array}ight|=34$ $\Rightarrow \quad 4\left|\begin{array}{cc}-1 & 1 \ 9 & p\end{array}ight|-5\left|\begin{array}{cc}0 & 1 \ 3 & p\end{array}ight|+1\left|\begin{array}{cc}0 & -1 \ 3 & 9\end{array}ight|=34$ $\Rightarrow \quad 4(-p-9)-5(-3)+1(3)=34$ $\Rightarrow \quad-4 p-36+15+3=34$ $\Rightarrow \quad 4 p=-36+18-34$ $\Rightarrow \quad p=-\frac{52}{4}=-13$

If the volume of parallelopiped with conterminus edges $4 \hat{\mathbf{i}}+5…

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