Volume of the parallelepiped from vectors

Formula for volume of the parallelepiped

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

$|a\cdot(b\times c)|$

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where the given vectors are $a\langle{a_1},a_2,a_3\rangle$, $b\langle{b_1},b_2,b_3\rangle$ and $c\langle{c_1},c_2,c_3\rangle$. $b\times c$ is the cross product of $b$ and $c$, and we’ll find it using the $3\times 3$ matrix

$\begin{vmatrix}\bold i&\bold j&\bold k\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}=\bold i\begin{vmatrix}b_2&b_3\\c_2&c_3\end{vmatrix}-\bold j\begin{vmatrix}b_1&b_3\\c_1&c_3\end{vmatrix}+\bold k\begin{vmatrix}b_1&b_2\\c_1&c_2\end{vmatrix}$

$=\bold i(b_2c_3-b_3c_2)-\bold j(b_1c_3-b_3c_1)+\bold k(b_1c_2-b_2c_1)$

We’ll convert the result of the cross product into standard vector form, and then take the dot product of $a\langle{a_1},a_2,a_3\rangle$ and the vector result of $b\times c$. The final answer is the value of the scalar triple product, which is the volume of the parallelepiped.

Using vectors that define the parallelepiped to find its volume

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Finding volume of the parallelepiped given by the vectors

Example

Find the volume of the parallelepiped given by the vectors.

$a\langle2,-1,3\rangle$

$b\langle3,2,-4\rangle$

$c\langle-2,0,1\rangle$

We’ll start by taking the cross product of $b$ and $c$.

$b\times c=\begin{vmatrix}\bold i&\bold j&\bold k \\ 3 & 2 & -4 \\ -2 & 0 & 1\end{vmatrix}$

$b\times c=\bold i\begin{vmatrix}2 & -4\\ 0 & 1\end{vmatrix}-\bold j\begin{vmatrix}3 & -4\\ -2 & 1\end{vmatrix}+\bold k\begin{vmatrix}3 & 2\\ -2 & 0\end{vmatrix}$

$b\times c=\left[(2)(1)-(-4)(0)\right]\bold i-\left[(3)(1)-(-4)(-2)\right]\bold j+\left[(3)(0)-(2)(-2)\right]\bold k$

$b\times c=(2+0)\bold i-(3-8)\bold j+(0+4)\bold k$

$b\times c=2\bold i+5\bold j+4\bold k$

$b\times c=\langle2,5,4\rangle$

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

Now we’ll take the dot product of $a\langle2,-1,3\rangle$ and $b\times c=\langle2,5,4\rangle$.

$|a\cdot(b\times c)|=(2)(2)+(-1)(5)+(3)(4)$

$|a\cdot(b\times c)|=4-5+12$

$|a\cdot(b\times c)|=11$

The volume of the parallelepiped is $11$.

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