Question

If the volume of the parallelepiped with $\left( \left( \overrightarrow{a}\times \overrightarrow{b} \right),\left( \overrightarrow{b}\times \overrightarrow{c} \right),\left( \overrightarrow{c}\times \overrightarrow{a} \right) \right)$ as coterminous edge is 8 cubic units, then the volume of the parallelepiped with $\left( \left( \overrightarrow{a}\times \overrightarrow{b} \right)\times \left( \overrightarrow{b}\times \overrightarrow{c} \right),\left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right),\left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right)$ as coterminous edges is?
(a) 8 cubic units
(b) 512 cubic units
(c) 64 cubic units
(d) 24 cubic units

Answer 1

Use the formula for the volume of a parallelepiped with coterminous edges as $\overrightarrow{p}$, $\overrightarrow{q}$ and $\overrightarrow{r}$ given by the scalar triple product as Volume = $\left| \overrightarrow{p}.\left( \overrightarrow{q}\times \overrightarrow{r} \right) \right|=\left| \overrightarrow{q}.\left( \overrightarrow{r}\times \overrightarrow{p} \right) \right|=\left| \overrightarrow{r}.\left( \overrightarrow{q}\times \overrightarrow{p} \right) \right|$. Find the volume expression for the parallelepiped with coterminous edge as $\left( \left( \overrightarrow{a}\times \overrightarrow{b} \right),\left( \overrightarrow{b}\times \overrightarrow{c} \right),\left( \overrightarrow{c}\times \overrightarrow{a} \right) \right)$ and equate it with 8. Now, find the volume expression with the coterminous edge as $\left( \left( \overrightarrow{a}\times \overrightarrow{b} \right)\times \left( \overrightarrow{b}\times \overrightarrow{c} \right),\left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right),\left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right)$ and relate it with former expression to get the answer.

Complete step-by-step solution:
Here we have been provided with the volume of a parallelepiped having its coterminous edge as $\left( \left( \overrightarrow{a}\times \overrightarrow{b} \right),\left( \overrightarrow{b}\times \overrightarrow{c} \right),\left( \overrightarrow{c}\times \overrightarrow{a} \right) \right)$. We are asked to find the volume for the parallelepiped whose coterminous edge is $\left( \left( \overrightarrow{a}\times \overrightarrow{b} \right)\times \left( \overrightarrow{b}\times \overrightarrow{c} \right),\left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right),\left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right)$.

Now, a parallelepiped having its coterminous edge as $\overrightarrow{p}$, $\overrightarrow{q}$ and $\overrightarrow{r}$ given by the scalar triple product as Volume = $\left| \overrightarrow{p}.\left( \overrightarrow{q}\times \overrightarrow{r} \right) \right|=\left| \overrightarrow{q}.\left( \overrightarrow{r}\times \overrightarrow{p} \right) \right|=\left| \overrightarrow{r}.\left( \overrightarrow{q}\times \overrightarrow{p} \right) \right|$. So we have the volume relation of the parallelepiped with coterminous edge $\left( \left( \overrightarrow{a}\times \overrightarrow{b} \right),\left( \overrightarrow{b}\times \overrightarrow{c} \right),\left( \overrightarrow{c}\times \overrightarrow{a} \right) \right)$ given as,
$\Rightarrow \left| \left( \overrightarrow{a}\times \overrightarrow{b} \right).\left( \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right) \right|=8$ …….. (1)
Using the similar scalar triple product formula for the volume (V) of the parallelepiped having coterminous edge $\left( \left( \overrightarrow{a}\times \overrightarrow{b} \right)\times \left( \overrightarrow{b}\times \overrightarrow{c} \right),\left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right),\left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right)$ can be given as,

& \Rightarrow V=\left| \left[ \left( \overrightarrow{a}\times \overrightarrow{b} \right)\times \left( \overrightarrow{b}\times \overrightarrow{c} \right) \right].\left[ \left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\}\times \left\\{ \left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right\\} \right] \right| \\\ & \Rightarrow V=\left| \left( \overrightarrow{a}\times \overrightarrow{b} \right).\left[ \left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\}\times \left\\{ \left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right\\} \right]\times \left( \overrightarrow{b}\times \overrightarrow{c} \right).\left[ \left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\}\times \left\\{ \left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right\\} \right] \right| \\\ \end{aligned}$$ Now, in the first part if we further break the terms then we will get $$\left( \overrightarrow{a}\times \overrightarrow{b} \right).\left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\}\times \left( \overrightarrow{a}\times \overrightarrow{b} \right).\left\\{ \left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right\\}$$. Here in the expression $$\left( \overrightarrow{a}\times \overrightarrow{b} \right).\left\\{ \left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right\\}$$ we can see that $$\left\\{ \left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right\\}$$ will be perpendicular to the vector $$\left( \overrightarrow{a}\times \overrightarrow{b} \right)$$ and therefore the final dot product will be equal to 0 as we have $\cos {{90}^{\circ }}=0$, similarly in the second term we will have $$\left( \overrightarrow{b}\times \overrightarrow{c} \right).\left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\}$$ equal to 0. Therefore the volume expression gets reduced to, $$\Rightarrow V=\left| \left( \overrightarrow{a}\times \overrightarrow{b} \right).\left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\}\times \left( \overrightarrow{b}\times \overrightarrow{c} \right).\left\\{ \left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right\\} \right|$$ Using the formula $\overrightarrow{p}.\left( \overrightarrow{q}\times \overrightarrow{r} \right)=\overrightarrow{q}.\left( \overrightarrow{r}\times \overrightarrow{p} \right)=\overrightarrow{r}.\left( \overrightarrow{q}\times \overrightarrow{p} \right)$ we can write $$\left( \overrightarrow{b}\times \overrightarrow{c} \right).\left\\{ \left( \overrightarrow{c}\times \overrightarrow{a} \right)\times \left( \overrightarrow{a}\times \overrightarrow{b} \right) \right\\}=\left( \overrightarrow{a}\times \overrightarrow{b} \right).\left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\}$$, so we get, $$\Rightarrow V=\left| \left( \overrightarrow{a}\times \overrightarrow{b} \right).\left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\}\times \left( \overrightarrow{a}\times \overrightarrow{b} \right).\left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\} \right|$$ We know that $\left| \overrightarrow{k}\times \overrightarrow{k} \right|={{\left| k \right|}^{2}}$ so we get, $$\Rightarrow V={{\left| \left( \overrightarrow{a}\times \overrightarrow{b} \right).\left\\{ \left( \overrightarrow{b}\times \overrightarrow{c} \right)\times \left( \overrightarrow{c}\times \overrightarrow{a} \right) \right\\} \right|}^{2}}$$ Using relation (1) we get, $$\begin{aligned} & \Rightarrow V={{\left| 8 \right|}^{2}} \\\ & \therefore V=64 \\\ \end{aligned}$$ Therefore the volume of the required parallelepiped is 64 cubic units. Hence, option (c) is the correct answer. **Note:** The formula of the volume of a parallelepiped is nothing but the formula we generally use for the volume calculation of any 3 – D shape given as Base area $\times $ Height. In the above solution it looks lengthy because we had to use the vector form and the coterminous edge expressions were lengthy. Remember certain useful concepts related to the dot product and cross product.