If $|\vec{a}|=2 \sqrt{2}$ and $|\vec{b}|=3$ and angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$. If a parallelogram is constructed with adjacent sides $\vec{p}=2 \vec{a}-3 \vec{b}$ and $\vec{q}=\vec{a}+\vec{b}$ then the product of length of both the diagonals is :

Position vector of P and Q are $\hat{\imath}+3 \hat{\jmath}-7 \hat{k}$ and $5 \hat{\imath}-2 \hat{\jmath}+4 \hat{k}$ respectively. Then the cosine of the angle between $\overrightarrow{P Q}$ and y -axis is

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $a \neq 0$ and $\vec{a} \times \vec{b}=2(\vec{a} \times \vec{c}),|\vec{a}|=|\vec{c}|=1,|\vec{b}|=4$ and $|\vec{b} \times \vec{c}|=\sqrt{15}$ if $\vec{b}-2 \vec{c}=\lambda \vec{a}$ then $\lambda^2$ equals :

A line $L_1$ passing through the point A with position vector $\vec{a}=4 \hat{i}+2 \hat{j}+2 \hat{k}$ is parallel to the vector $\vec{b}=2 \hat{i}+3 \hat{j}+6 \hat{k}$. The length of the perpendicular drawn from a point P with position vector $\vec{p}=\hat{i}+2 \hat{j}+3 \hat{k}$ to $L_1$ is