If $\vec{a} = \langle \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \rangle$,then what is its magnitude?

If the position vectors of points $A, B, C$ are respectively $\hat{i}, \hat{j}, \hat{k}$ and $\vec{AB} = \vec{CX}$,then the position vector of point $X$ is:

Let $|\vec{a}| = |\vec{b}| = |\vec{a} - \vec{b}| = 1$,then the angle between $\vec{a}$ and $\vec{b}$ is:

The direction cosines of the vector $3\hat{i} - 4\hat{j} + 5\hat{k}$ are

If the vector $a = 2 \hat{i} + 3 \hat{j} + 6 \hat{k}$ and $b$ are collinear and $|b| = 21$,then $b$ is equal to:

If $\vec{a}$ and $\vec{b}$ are non-collinear vectors,then the value of $\alpha$ for which the vectors $\vec{u} = (\alpha - 2)\vec{a} + \vec{b}$ and $\vec{v} = (2 + 3\alpha)\vec{a} - 3\vec{b}$ are collinear is: