The sum of the lengths of projections of $p \hat{i} + q \hat{j} + r \hat{k}$ on the coordinate axes, where $p = 4, q = -5, r = 7$, is: (in $\text{ units}$)

Let $\vec{a}=x \hat{i}+y \hat{j}+z \hat{k}$ and $x=2 y$. If $|\vec{a}|=5 \sqrt{2}$ and $\vec{a}$ makes an angle of $135^{\circ}$ with the $z$-axis,then $\vec{a}=$

Let $A, B$ and $C$ be three points on a circle of radius $R$. If $O$ is the centre of the circle and $\angle AOB = 45^{\circ}, \angle BOC = 45^{\circ}$,then the resultant of $\vec{OA}, \vec{OB}$ and $\vec{OC}$ has magnitude

If $\vec{a} = 4\hat{i} + 3\hat{j}$ and $\vec{b} = 2\hat{i} + \lambda\hat{j}$ are parallel vectors,then the value of $\lambda$ is:

If $a = 2i + 5j$ and $b = 2i - j,$ then the unit vector along $a + b$ will be

The unit vector in the direction of $\vec{x} = (2, 3, \sqrt{3})$ is . . . . . . .