Assertion (A) For the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}$ and $\mathbf{r}=\mathbf{p}+s \mathbf{q}$, if $(\mathbf{a}-\mathbf{p}) \cdot(\mathbf{b} \times \mathbf{q}) \neq 0$, then the two lines are coplanar.

Reason $(\mathrm{R})|(\mathbf{a}-\mathbf{p}) \cdot(\mathbf{b} \times \mathbf{q})|$ is $|\mathbf{b} \times \mathbf{q}|$ times the shortest distance between the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}$ and $\mathbf{r}=\mathbf{p}+s \mathbf{q}$.

The locus of a point at which the line joining the points $(-3,1,2),(1,-2,4)$ subtends a right angle, is

If $A(1,2,3), B(2,3,-1), C(3,-1,-2)$ are the vertices of a $\triangle A B C$, then the direction ratios of the bisector of $\angle A B C$ are

Let $A=(2,0,-1), B=(1,-2,0), C=(1,2,-1)$ and $D=(0,-1,-2)$ be four points.

If $\theta$ is the acute angle between the plane determined by $A, B, C$ and the plane determined by $A, C, D$, then $\tan \theta=$