The value of $m \in R$,when the angle between the vectors $\bar{p} = m \hat{i} - 6 \hat{j} + 3 \hat{k}$ and $\bar{q} = \hat{i} + 2 \hat{j} + 2m \hat{k}$ is obtuse,is

If $|\vec{a}|=1, |\vec{b}|=2, |\vec{a}-\vec{b}|^2+|\vec{a}+2\vec{b}|^2=20$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is:

Let $\bar{a} = 2\bar{i} - \bar{j} + \bar{k}$,$\bar{b} = \bar{i} + 2\bar{j} - \bar{k}$,and $\bar{c} = \bar{i} + \bar{j} - 2\bar{k}$ be three vectors. $A$ vector $\bar{r}$ in the plane of $\bar{b}$ and $\bar{c}$ has a projection of magnitude $\sqrt{\frac{2}{3}}$ on the vector $\bar{a}$. Find $\bar{r}$.

If $\vec{a}$ and $\vec{b}$ are two vectors such that $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} < 0$ and $|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$,then the angle between the vectors $\vec{a}$ and $\vec{b}$ is

If $\vec{a} = \hat{i} + 3\hat{j} - 2\hat{k}$ and $\vec{b} = 4\hat{i} - 2\hat{j} + 4\hat{k}$,then $(2\vec{a} + \vec{b}) \cdot (\vec{a} - 2\vec{b}) = \dots$

If two vectors $\vec{a}$ and $\vec{b}$ which are perpendicular to each other are such that $|\vec{a}|=8$ and $|\vec{b}|=3$, then $|\vec{a}-2\vec{b}|=$