Question

what is the volume of the parallelepiped determined by the vectors ( a = langle 1, - 2, 3
angle ), ( b = langle 2, 4, 2
angle ), and ( c = langle 2, 1, 4
angle )?
4 cubic units
20 cubic units
24 cubic units
56 cubic units

Explanation:

Step1: Recall the formula for the volume of a parallelepiped

The volume \(V\) of the parallelepiped determined by vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is given by \(V=\vert\vec{a}\cdot(\vec{b}\times\vec{c})\vert\).
First, find the cross - product \(\vec{b}\times\vec{c}\).
If \(\vec{b}=\langle2,4,2
angle\) and \(\vec{c}=\langle2,1,4
angle\), then \(\vec{b}\times\vec{c}=

$$\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\2&4&2\\2&1&4\end{vmatrix}$$

\)

$$
LATEXBLOCK1
$$

Step2: Find the dot - product \(\vec{a}\cdot(\vec{b}\times\vec{c})\)

Given \(\vec{a}=\langle1,-2,3
angle\) and \(\vec{b}\times\vec{c}=\langle14,-4,-6
angle\)
\(\vec{a}\cdot(\vec{b}\times\vec{c})=(1\times14)+(-2\times(-4))+(3\times(-6))\)

$$
LATEXBLOCK2
$$

Step3: Find the volume

Since \(V = \vert\vec{a}\cdot(\vec{b}\times\vec{c})\vert\), and \(\vec{a}\cdot(\vec{b}\times\vec{c}) = 4\), then \(V=\vert4\vert=4\)

Answer:

4 cubic units