Let $\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}$ be a vector such that $\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$ and $\vec{a} \cdot \vec{b}=3$. Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is :-

If $\hat{a}, \hat{b},$ and $\hat{c}$ are unit vectors satisfying $\hat{a} - \sqrt{3}\hat{b} + \hat{c} = \vec{0},$ then the angle between the vectors $\hat{a}$ and $\hat{c}$ is

For any two vectors $\vec{a}$ and $\vec{b}$,$|\vec{a}| |\vec{b}|$ . . . . . . $|\vec{a} \cdot \vec{b}|$.

Let $m$ be the unit vector orthogonal to the vector $\hat{i}-\hat{j}+\hat{k}$ and coplanar with the vectors $2 \hat{i}+\hat{j}$ and $\hat{j}-\hat{k}$. If $a=\hat{i}-\hat{k}$,then the length of the perpendicular from the origin to the plane $r \cdot m=a \cdot m$ is

Let $\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}$,$\vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}$,and $\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \times \vec{a}$ and $\vec{r} \cdot (\vec{b}-\vec{c})=0$,then $\frac{|593 \vec{r}+67 \vec{a}|^2}{(593)^2}$ is equal to:

If $\overline{a}+\overline{b}+\overline{c}=\overline{0}$ with $|\overline{a}|=3, |\overline{b}|=5$ and $|\overline{c}|=7$,then the angle between $\overline{a}$ and $\overline{b}$ is: