Ncert Exercises & Solutions

Question 7. 5. Show that $\vec{a} . (\vec{b} x \vec{c})$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $\vec{a}, \vec{b} and \overset{⇀}{c}$ .

A parallelepiped with origin O and sides a, b, and c is shown in the following figure.

Volume of the given parallelepiped = abc

$\vec{OC} = \vec{a}$

$\vec{OB} = \vec{b}$

$\vec{OC} = \vec{c}$

Let

$\hat{n}$ be a unit vector perpendicular to both b and c. Hence,

$\hat{n}$ and a have the same
direction.

$∴ \vec{b} \times \vec{c} = bcsinθ \hat{n}$

$= bc \sin 90 ° \hat{n}$

$= bc \hat{n}$

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

$= a \cdot (bc \hat{n})$

$= abc cosθ \hat{n}$

$= abc \cos 0 °$

$= abc$

$= Volume of the parallelpiped$

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