If $\bar{a}, \bar{b}, \bar{c}$ are the position vectors of the points $A(1,3,0), B(2,5,0), C(4,2,0)$ respectively and $\bar{c}=t_{1} \bar{a}+t_{2} \bar{b}$,then the value of $t_{1} t_{2}$ is:

$A$ vector of magnitude $14$ lies in the $xy-$ plane and makes an angle of $60^\circ$ with the $x-$ axis. The components of the vector in the direction of the $x-$ axis and $y-$ axis are:

If $a = \hat{i} + 4 \hat{j}$, $b = 2 \hat{i} - 2 \hat{j}$, and $c = 5 \hat{i} + 9 \hat{j}$, then $c$ is equal to:

The components of a vector $\vec{a}$ with respect to a rectangular Cartesian system are $2p$ and $1$. The system is rotated about the origin through a certain angle in the counter-clockwise direction. If the components of $\vec{a}$ with respect to the new system are $p + 1$ and $1$,then:

The unit vector $\vec{r}$ which satisfies $\vec{r} \times \vec{b} = \vec{r} \times \vec{c}$,where $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$ and $\vec{c} = 3\hat{i} + 2\hat{k}$,is:

In a regular hexagon $ABCDEF$,$\overrightarrow{AB}=\vec{a}$ and $\overrightarrow{BC}=\vec{b}$,then $\overrightarrow{FA}=$