Question

Let a = (3,1,2) b = (- 1,2,1) and c = (2,1, a) Set the parameter so that the volume of the parallelepiped is 24​

Answer 1

Either
`c = (2,\, 1,\, (-13) / 7)` or
`c = (2,\, 1,\, 5)`.

Explanation:

The volume of a parallelepiped with sides
`\vec{a}`,
`\vec{b}`, and
`\vec{c}` is the absolute value of
`\left(\vec{a} * \vec{b}\right)\, \vec{c}` (a cross product between the first two sides followed by a dot product with the third side.)

In this example:

`\begin{aligned} &\vec{a} * \vec{b} \\ =\; & \begin{bmatrix}3 \\ 1 \\ 2\end{bmatrix} * \begin{bmatrix}-1 \\ 2 \\ 1\end{bmatrix} \\ =\; & \begin{bmatrix}1 * 1 - 2 * 2 \\ 2 * (-1) - 3 * 1 \\ 3 *2 - 1 * (-1) \end{bmatrix} \\ =\; & \begin{bmatrix} -3 \\ -5 \\ 7 \end{bmatrix}\end{aligned}`.

Given that
`\vec{c} = (2,\, 1,\, m)` for some unknown parameter
`m`:

`\begin{aligned}& \left(\vec{a} * \vec{b}\right)\, \vec{c} \\ =\; & \begin{bmatrix}-3 \\ -5 \\ 7\end{bmatrix}\, \begin{bmatrix}2 \\ 1 \\ m\end{bmatrix} \\ =\; & (-3) * 2 + (-5) * 1 + 7 \, m \\ =\; & -11 + 7\, m\end{aligned}`.

The volume of this parallelepiped would be the absolute value
`|-11 + 7\, m|`. For this volume to be equal to
`24`, either
`(-11 + 7\, m) = (-24)`, such that
`m = (-13) / 7`, or
`(-11 + 7\, m) = 24`, such that
`m = 5`.