The altitude of the parallelopiped, whose coterminous edges are the vectors $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - \hat{k}$, $\vec{c} = \hat{i} + \hat{j} + 3\hat{k}$, where $\vec{a}$, $\vec{b}$ are the sides of the base of parallelopiped, is

Let ABCD be a quadrilateral with $\overline{AB} = \vec{a}$, $\overline{AD} = \vec{b}$ and $\overline{AC} = 3\vec{a} + 2\vec{b}$. If its area is $\alpha$ times the area of the parallelogram with AB, AD as adjacent sides, then the value of $\alpha$ is equal to

Let $\vec{a} = \hat{i} + 2\hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $|\vec{c} - \vec{a}| = 2\sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $60^\circ$, then $|(\vec{a} \times \vec{b}) \times \vec{c}|$ is equal to

If $\bar{a}$ and $\bar{b}$ are unit vectors and $\theta$ ($0 < \theta < \pi$) is the angle between them, then the value of $\dfrac{|\bar{a} + \bar{b}|}{|\bar{a} - \bar{b}|}$ is equal to...