How to find the volume of the parallelepiped from its adjacent edges

Formulas for volume of the parallelepiped

If we need to find the volume of a parallelepiped and we’re given three adjacent edges of it, all we have to do is find the scalar triple product of the three vectors that define the edges:

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$|\vec{PS}\cdot(\vec{PQ}\times \vec{PR})|$

where $\vec{PS}$, $\vec{PQ}$ and $\vec{PR}$ are the three adjacent edges.

First we’ll find the vectors $\vec{PQ}$, $\vec{PR}$ and $\vec{PS}$, then we’ll find the cross product $\vec{PQ}\times \vec{PR}$ using the $3\times 3$ matrix

$\begin{vmatrix}\bold i&\bold j&\bold k\\PQ_1&PQ_2&PQ_3\\PR_1&PR_2&PR_3\end{vmatrix}=\bold i\begin{vmatrix}PQ_2&PQ_3\\PR_2&PR_3\end{vmatrix}-\bold j\begin{vmatrix}PQ_1&PQ_3\\PR_1&PR_3\end{vmatrix}+\bold k\begin{vmatrix}PQ_1&PQ_2\\PR_1&PR_2\end{vmatrix}$

$=(PQ_2PR_3-PQ_3PR_2)\bold i-(PQ_1PR_3-PQ_3PR_1)\bold j+(PQ_1PR_2-PQ_2PR_1)\bold k$

We’ll convert the result of the cross product into standard vector form, and then take the dot product of $\vec{PS}\langle{PS_1},PS_2,PS_3\rangle$ and the vector result of $\vec{PQ}\times\vec{PR}$. The final answer is the value of the scalar triple product, which is the volume of the parallelepiped.

How to find the volume of a parallelepiped, given three vectors that define its edges

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Building three vectors from four points, then using the vectors to find the parallelepiped’s volume

Example

Find the volume of the parallelepiped given by the adjacent edges $\vec{PQ}$, $\vec{PR}$ and $\vec{PS}$.

$P(5,1,-2)$

$Q(0,-1,3)$

$R(3,2,-4)$

$S(1,-2,0)$

We need to start by using the four points to find the vectors $\vec{PQ}$, $\vec{PR}$ and $\vec{PS}$, since these are the three adjacent edges of the parallelepiped.

$\vec{PQ}=\langle0-5,-1-1,3-(-2)\rangle$

$\vec{PQ}=\langle-5,-2,5\rangle$

and

$\vec{PR}=\langle3-5,2-1,-4-(-2)\rangle$

$\vec{PR}=\langle-2,1,-2\rangle$

and

$\vec{PS}=\langle1-5,-2-1,0-(-2)\rangle$

$\vec{PS}=\langle-4,-3,2\rangle$

The final answer is the value of the scalar triple product, which is the volume of the parallelepiped.

Now we need to take the cross product of $\vec{PQ}$ and $\vec{PR}$.

$\vec{PQ}\times\vec{PR}=\begin{vmatrix}\bold i&\bold j&\bold k\\-5&-2&5\\-2&1&-2\end{vmatrix}$

$\vec{PQ}\times\vec{PR}=\bold i\begin{vmatrix}-2&5\\1&-2\end{vmatrix}-\bold j\begin{vmatrix}-5&5\\-2&-2\end{vmatrix}+\bold k\begin{vmatrix}-5&-2\\-2&1\end{vmatrix}$

$\vec{PQ}\times\vec{PR}=\left[(-2)(-2)-(5)(1)\right]\bold i-\left[(-5)(-2)-(5)(-2)\right]\bold j+\left[(-5)(1)-(-2)(-2)\right]\bold k$

$\vec{PQ}\times\vec{PR}=(4-5)\bold i-(10+10)\bold j+(-5-4)\bold k$

$\vec{PQ}\times\vec{PR}=-\bold i-20\bold j-9\bold k$

$\vec{PQ}\times\vec{PR}=\langle-1,-20,-9\rangle$

Taking the dot product of $\vec{PS}=\langle-4,-3,2\rangle$ and $\vec{PQ}\times\vec{PR}=\langle-1,-20,-9\rangle$, we get

$\left|\vec{PS}\cdot\left(\vec{PQ}\times\vec{PR}\right)\right|=(-4)(-1)+(-3)(-20)+(2)(-9)$

$\left|\vec{PS}\cdot\left(\vec{PQ}\times\vec{PR}\right)\right|=4+60-18$

$\left|\vec{PS}\cdot\left(\vec{PQ}\times\vec{PR}\right)\right|=46$

The volume of the parallelepiped is $46$.

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