$ABCDEF$ is a regular hexagon whose centre is $O$. Then,$\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF}$ is equal to (in $vec{AO}$)

In $\triangle OAC$,if $B$ is the mid-point of side $AC$ and $\vec{OA}=\vec{a}, \vec{OB}=\vec{b}$,then $\vec{OC}$ is equal to

If the vectors $\vec{a} = \lambda \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{b} = \sqrt{\lambda} \hat{i} + \sqrt{13} \hat{j}$ have the same magnitude,then the value of $\lambda$ is:

If the points with position vectors $\alpha \hat{i} + 10 \hat{j} + 13 \hat{k}$,$6 \hat{i} + 11 \hat{j} + 11 \hat{k}$,and $\frac{9}{2} \hat{i} + \beta \hat{j} - 8 \hat{k}$ are collinear,then $(19 \alpha - 6 \beta)^2$ is equal to $...........$.

If $\vec{a} + 5\vec{b} = \vec{c}$ and $\vec{a} - 7\vec{b} = 2\vec{c}$,then which of the following is true?

If $ABCD$ is a parallelogram,$\overrightarrow{AB} = 2i + 4j - 5k$ and $\overrightarrow{AD} = i + 2j + 3k,$ then the unit vector in the direction of $\overrightarrow{BD}$ is