The shortest distance between the lines $r = 3i + 5j + 7k + \lambda(i + 2j + k)$ and $r = -i - j - k + \mu(7i - 6j + k)$ is

If the adjacent sides of a rectangle are $\bar{a}=5\bar{m}-3\bar{n}$,$\bar{b}=-\bar{m}-2\bar{n}$ and the adjacent sides of another rectangle are $\bar{c}=-4\bar{m}-\bar{n}$,$\bar{d}=-\bar{m}+\bar{n}$,then the angle between the vectors $\bar{x}=\frac{\bar{a}+\bar{c}+\bar{d}}{3}$ and $\bar{y}=\frac{\bar{c}+\bar{d}}{5}$ is

If $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ and $|\vec{a} \times \vec{b}| = |\vec{a} \cdot \vec{b}|$,then $\theta$ is equal to

If the vertices $A, B, C$ of a triangle $ABC$ are $(1,2,3), (-1,0,0), (0,1,2)$ respectively,then find $\angle ABC$. $[\angle ABC \text{ is the angle between the vectors } \overrightarrow{BA} \text{ and } \overrightarrow{BC}]$.

If $\bar{a}=\hat{i}+2 \hat{j}+\hat{k}$,$\bar{b}=\hat{i}-\hat{j}+\hat{k}$,and $\bar{c}=\hat{i}+\hat{j}-\hat{k}$,then a vector in the plane of $\bar{a}$ and $\bar{b}$,whose projection on $\bar{c}$ is $\frac{1}{\sqrt{3}}$,is

The magnitude of the projection of vector $\vec{a} = -\hat{i} + 2\hat{j} - \hat{k}$ on the vector $\vec{b} = \hat{i} + 2\hat{j} + 2\hat{k}$ is . . . . . . .