Question

Find the volume of a parallelepiped whose defining vectors are

Short answer

The volume is : 226 cubic units

Step-by-step solution

Step 1. given vectors

Which can be written as :

$$\begin{gathered} \overrightarrow{A}=\hat{i}+0\hat{j}+6\hat{k} \\ \overrightarrow{B}=2\hat{i}+3\hat{j}-\hat{k} \\ \overrightarrow{C}=8\hat{i}-5\hat{j}+6\hat{k} \end{gathered}$$

Step 2. Volume of parallelopiped 

As given in question volume of parallelopiped is given by :

$V=\left|(A\times B).C\right|$

So , Using this

$$\begin{gathered} \overrightarrow{A}=\hat{i}+0\hat{j}+6\hat{k} \\ \overrightarrow{B}=2\hat{i}+3\hat{j}-8\hat{k} \\ \overrightarrow{C}=8\hat{i}-5\hat{j}+6\hat{k} \\ V=\left|(A\times B).C\right| \\ V=\left|\right((\hat{i}+0\hat{j}+6\hat{k})\times (2\hat{i}+3\hat{j}-8\hat{k})).(8\hat{i}-5\hat{j}+6\hat{k}\left)\right| \\ V=\left|\right(-18\hat{i}+20\hat{j}+3\hat{k})\times (8\hat{i}-5\hat{j}+6\hat{k}\left)\right| \\ V=|-144-100+18| \\ V=226cubicunits \end{gathered}$$