Relative to a fixed origin $$\(O\)$$, - the point $$\(A\)$$ has position vector $$\[ \mathbf{i}-4 \mathbf{j}+3 \mathbf{k} \]$$ - the point $$\(B\)$$ has position vector $$\(5 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}\)$$ - the point $$\(C\)$$ has position vector $$\(3 \mathbf{i}+p \mathbf{j}-\mathbf{k}\)$$ where $$\(p\)$$ is a constant. The line $$\(l\)$$ passes through $$\(A\)$$ and $$\(B\)$$. (a) Find a vector equation for the line $$\(l\)$$ (3) Given that $$\(\overrightarrow{A C}\)$$ is perpendicular to $$\(l\)$$ (b) find the value of $$\(p\)$$ (3) (c) Hence find the area of triangle $$\(A B C\)$$, giving your answer as a surd in simplest form. (3)

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