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 \(\mathbf{OP}\).

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

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