Problem

Relative to a fixed origin $O$,

• the point $A$ has position vector $\mathbf{i}-4\mathbf{j}+3\mathbf{k}$
• the point $B$ has position vector $5\mathbf{i}+3\mathbf{j}-2\mathbf{k}$
• the point $C$ has position vector $3\mathbf{i}+p\mathbf{j}-\mathbf{k}$

where $p$ is a constant.

The line $l$ passes through $A$ and $B$.

(a) Find a vector equation for the line $l$

Given that $\overrightarrow{AC}$ is perpendicular to $l$

(b) find the value of $p$

(c) Hence find the area of triangle $ABC$, giving your answer as a surd in simplest form.