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