If $\overrightarrow {AO} + \overrightarrow {OB} = \overrightarrow {BO} + \overrightarrow {OC} ,$ then $A, B, C$ form

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:

If $\overline{e}_1, \overline{e}_2$ and $\overline{e}_1+\overline{e}_2$ are unit vectors,then the angle between $\overline{e}_1$ and $\overline{e}_2$ is (in $^{\circ}$)

If the vectors $3\,i + 2\,j - k$ and $6\,i - 4xj + yk$ are parallel,then the value of $x$ and $y$ will be

If $|\vec{c}|^2 = 60$ and $\vec{c} \times (\hat{i} + 2\hat{j} + 5\hat{k}) = \vec{0}$,then a value of $\vec{c} \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k})$ is

Let $u, v, w$ be vectors such that $u + v + w = 0$. If $|u| = 3, |v| = 4,$ and $|w| = 5,$ find the value of $u \cdot v + v \cdot w + w \cdot u$.