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 $\vec a = \hat i - \hat j,$ $\vec b = \hat i + \hat j + \hat k$ and $\vec c$ be a vector such that $\vec a \times \vec c + \vec b = 0$ and $\vec a \cdot \vec c = 4$,then ${\left| {\vec c} \right|^2}$ is equal to

Let $\overline{u}, \overline{v}$ and $\overline{w}$ be the vectors such that $|\overline{u}|=1, |\overline{v}|=2, |\overline{w}|=3$. If the projection of $\overline{v}$ along $\overline{u}$ is equal to that of $\overline{w}$ along $\overline{u}$ and $\overline{v}, \overline{w}$ are perpendicular to each other,then $|\overline{u}-\overline{v}+\overline{w}|$ is equal to

If $|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}=144$ and $|\vec{a}|=4$,then the value of $|\vec{b}|$ is

If $a, b, c$ are three mutually perpendicular vectors such that the magnitudes of $b$ and $c$ are $1/2$ times and $\sqrt{3}/2$ times that of $a$, respectively, then the angle between the vectors $a+b+c$ and $b$ is

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. $A$ vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$,whose projection on $\vec{c}$ is $1/\sqrt{3}$,is given by: