If the angle between the vectors $2 \alpha^2 \hat{i} + 4 \alpha \hat{j} + \hat{k}$ and $7 \hat{i} - 2 \hat{j} + \alpha \hat{k}$ is obtuse,then

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b} = \hat{i} - \hat{j} + \hat{k}$,and $\vec{c} = \hat{i} + \hat{j} - \hat{k}$ be three vectors. If $\vec{v}$ is a vector in the plane of $\vec{a}$ and $\vec{b}$ such that the projection of $\vec{v}$ on $\vec{c}$ is $\frac{1}{\sqrt{3}}$,then $\vec{v} = $

If $\overrightarrow{a} \cdot \hat{i}=4$,then $(\overrightarrow{a} \times \hat{j}) \cdot(2 \hat{j}-3 \hat{k})$ is equal to

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$. Then $|\vec{a}+2\vec{b}|^{2}+|\vec{a}+2\vec{c}|^{2}$ is equal to

If $a=\hat{i}+\hat{j}+t \hat{k}$ and $b=\hat{i}+2 \hat{j}+3 \hat{k}$,then the values of $t$ for which $(a+b)$ and $(a-b)$ are perpendicular are:

If $\overline{a}, \overline{b}, \overline{c}$ are non-coplanar vectors and $\overline{p}=\frac{\overline{b} \times \overline{c}}{[\overline{a} \overline{b} \overline{c}]}, \overline{q}=\frac{\overline{c} \times \overline{a}}{[\overline{a} \overline{b} \overline{c}]}, \overline{r}=\frac{\overline{a} \times \overline{b}}{[\overline{a} \overline{b} \overline{c}]}, \quad$ then $2 \overline{a} \cdot \overline{p}+\overline{b} \cdot \overline{q}+\overline{c} \cdot \overline{r}=$