1.13.5 Vectors to solve problems

Relative to a fixed origin $O$ ,

the point $A$ has position vector $2i+j+4k2\mathbf{i} + \mathbf{j} + 4\mathbf{k}$
the point $B$ has position vector $5i−2j+k5\mathbf{i} - 2\mathbf{j} + \mathbf{k}$
the point $C$ has position vector $qi+3j+5kq\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}$

where $q$ is a constant.

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

a.

Find a vector equation for the line $l$ .

[2]

b.

Given that $AC⃗\vec{AC}$ is perpendicular to $l$ ,

find the value of $q$ .

[3]

c.

Hence find the area of triangle $A B C$ , giving your answer as a surd in simplest form.

[4]