Show that $a \cdot(b \times c)$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $a, b$ and $c$.

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Show that $a \cdot(b \times c)$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $a, b$ and $c$ .

Show that $a \cdot(b \times c)$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $a, b$ and $c$ .

Let the parallelepiped formed be:

where,

$\overrightarrow{O J}=\vec{a}, \overrightarrow{O L}=\vec{b}$ and $\overrightarrow{O K}=\vec{c}$

$\hat{n}$ is a unit vector along $\mathrm{OJ}$ and is perpendicular to the plane containing the vectors $\vec{b}$ and $\vec{c}$

Now, $\quad \vec{b} \times \vec{c}=\mathrm{bcsin} 90^{\circ} \hat{n}$

$=\mathrm{bc} \hat{n}$

$\vec{a} \cdot(\vec{b} \times \vec{c})=a \cdot(b c \hat{n})$

$=a b c \cos \theta \hat{n}$

$=a b c \cos 0^{\circ}$

$=a b c$

Volume of the parallelepiped.

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