Given the following simultaneous equations for vectors $x$ and $y$:
$(i) x + y = a$
$(ii) x \times y = b$
$(iii) x \cdot a = 1$
Then $x = ?, y = ?$

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=\sqrt{14}$,$|\vec{b}|=\sqrt{6}$ and $|\vec{a} \times \vec{b}|=\sqrt{48}$. Then $(\vec{a} \cdot \vec{b})^2$ is equal to $...........$.

For a triangle $ABC$ with vertices $A(1, 0, 0)$,$B(0, 1, 0)$,and $C(0, 0, 1)$,the angle $A = \dots$

For what value of $x$ is the angle between the vectors $\vec{a} = -3\hat{i} + x\hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + 2x\hat{j} + \hat{k}$ acute,and the angle between $\vec{b}$ and the $x$-axis lies between $\pi/2$ and $\pi$?

$ABCD$ is a parallelogram. The position vectors of $A$ and $C$ are respectively $3\hat{i} + 3\hat{j} + 5\hat{k}$ and $\hat{i} - 5\hat{j} - 5\hat{k}$. If $M$ is the midpoint of the diagonal $DB$,then the magnitude of the projection of $\vec{OM}$ on $\vec{OC}$,where $O$ is the origin,is

Let the vectors $\overline{a}, \overline{b}, \overline{c}$ be such that $|\overline{a}|=2, |\overline{b}|=4$ and $|\overline{c}|=4$. If the projection of $\overline{b}$ on $\overline{a}$ is equal to the projection of $\overline{c}$ on $\overline{a}$ and $\overline{b}$ is perpendicular to $\overline{c}$,then the value of $|\overline{a}+\overline{b}-\overline{c}|$ is equal to