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

If a unit vector lies in the $yz$-plane and makes angles of $30^\circ$ and $60^\circ$ with the positive $y$-axis and $z$-axis respectively,then its components along the coordinate axes are:

The value of $x$ if $x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector is:

Let $P, Q, R,$ and $S$ be points on a plane with position vectors $-2\hat{i} - \hat{j}$,$4\hat{i}$,$3\hat{i} + 3\hat{j}$,and $-3\hat{i} + 2\hat{j}$ respectively. What type of quadrilateral is $PQRS$?

Let $\vec{u}, \vec{v}, \vec{w}$ be vectors such that $\vec{u} + \vec{v} + \vec{w} = \vec{0}$. If $|\vec{u}| = 3$,$|\vec{v}| = 4$,and $|\vec{w}| = 5$,then $\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u}$ is:

If the vectors $x \hat{i}-3 \hat{j}+7 \hat{k}$ and $\hat{i}+y \hat{j}-z \hat{k}$ are collinear,then the value of $\frac{x y^2}{z}$ is equal to: