Find the angle $\theta$ between the vectors $\vec{a} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$.

$|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = \dots$

Let three vectors $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be such that $\overrightarrow{c}$ is coplanar with $\overrightarrow{a}$ and $\overrightarrow{b}$,$\overrightarrow{a} \cdot \overrightarrow{c} = 7$ and $\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$,where $\overrightarrow{a} = -\hat{i} + \hat{j} + \hat{k}$ and $\overrightarrow{b} = 2\hat{i} + \hat{k}$. Then the value of $2|\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}|^{2}$ is .........

Let $\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$ and $\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}$ be two vectors,such that $\vec{a} \times \vec{b}=-\hat{i}+9 \hat{j}+12 \hat{k}$. Then the projection of $\vec{b}-2 \vec{a}$ on $\vec{b}+\vec{a}$ is equal to.

If the position vectors of $P$ and $Q$ are $\hat{i}+2 \hat{j}-7 \hat{k}$ and $5 \hat{i}-3 \hat{j}+4 \hat{k}$ respectively,then the cosine of the angle between $\overrightarrow{PQ}$ and $z$-axis is

If $|\bar{a}|=2, |\bar{b}|=3$ and $\bar{a}, \bar{b}$ are mutually perpendicular vectors,then the area of the triangle whose vertices are $0, \bar{a}+2\bar{b}, \bar{a}-2\bar{b}$ is