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

Answer:

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

Step-by-step explanation:

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

In this example:

$\begin{aligned} &\vec{a} \times \vec{b} \\ =\; & \begin{bmatrix}3 \\ 1 \\ 2\end{bmatrix} \times \begin{bmatrix}-1 \\ 2 \\ 1\end{bmatrix} \\ =\; & \begin{bmatrix}1 \times 1 - 2 \times 2 \\ 2 \times (-1) - 3 \times 1 \\ 3 \times2 - 1 \times (-1) \end{bmatrix} \\ =\; & \begin{bmatrix} -3 \\ -5 \\ 7 \end{bmatrix}\end{aligned}$.

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

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

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 $m = 5$.