If the position vectors of the vertices $A, B, C$ of a triangle are $3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 5(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})$ respectively, then the magnitude of the altitude drawn from $A$ on to the side $B C$ is

If the vectors $2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $p \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ are coplanar, then the unit vector in the direction of the vector $9 p \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ is

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 $\mathbf{r}=\mathbf{p}+s \mathbf{q}$.

Let $\mathbf{a}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ and $\mathbf{b}$ be two perpendicular vectors in the $X O Y$-plane. A vector $\mathbf{c}$ in the same plane and having projections 1 and 2 respectively on $\mathbf{a}$ and $\mathbf{b}$ is