Question -9

The scalar triple product $[ \vec{u} \quad \vec{v} \quad \vec{w} ]$ can be written as $\vec{u} \cdot ( \vec{v} \times \vec{w} )$ . We have $\vec{v} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{w} = \hat{i} + 3\hat{k}$ . Therefore $\vec{p} = ( \vec{v} \times \vec{w} ) = 3\hat{i} - 7\hat{j} - \hat{k}$ . The maximum value of the scalar triple product is the maximum value of the dot product of the vectors $\vec{p}$ and $\vec{u}$ , which is nothing but the product of the magnitude of the two vectors. Also since $\vec{u}$ is a unit vector, $[ \vec{u} \quad \vec{v} \quad \vec{w} ]_{ \text{max} } = | \vec{p} | = \sqrt{59}$ .