The position vectors of points \(A\), \(B\) and \(C\), referred to an origin \(O\), are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively, where

\[\begin{align*} \mathbf{a} &= 3\mathbf{i} + 4 \mathbf{j} + 5\mathbf{k}, \\ \mathbf{b} &= 7\mathbf{i} - \mathbf{k}, \\ \mathbf{c} &= 5\mathbf{i} + 5 \mathbf{j}. \\ \end{align*}\]

\(P\) is a point which is equidistant from the lines \(OA\), \(OB\) and \(OC\). Write down expressions for the cosine of the angle between \(OP\) and \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively and hence deduce an expression for a unit vector in the direction of \(\mathbf{OP}\).

We want to find the cosine of an angle between vectors. What does this suggest?

We are also interested in the distance of \(P\) from the lines. Can you draw a diagram showing the point on \(OA\) closest to \(P\)?

Given that the line \(OP\) meets the plane \(ABC\) at \(Q\), find \(\mathbf{OQ}\).

We will need equations for both the line \(OP\) and the plane \(ABC\) to find the intersection point.

Can you write the equation of the plane in a helpful form?

Given that \(G\) is the centroid of the triangle \(ABC\), show that \(QG\) is parallel to the coordinate plane \(Oyz\).

Can we define the position vector of \(G\) in terms of the those of the three corners?

Which vectors are parallel to the coordinate plane \(Oyz\)?