Use a vector representing a section of a line segment, including the midpoint of a line segment.

Three unmanned aerial vehicles (UAVs) are positioned at points \(P\), \(Q\), and \(R\) in three-dimensional space, where \(P\) has position vector \(\vec{p} = 4\mathbf{i} + 7\mathbf{j} + 2\mathbf{k}\), \(Q\) has position vector \(\vec{q} = 10\mathbf{i} + 3\mathbf{j} + 8\mathbf{k}\), and \(R\) has position vector \(\vec{r} = 16\mathbf{i} - \mathbf{j} + 14\mathbf{k}\), where distances are measured in metres. a) Determine the position vector of point \(M\), the midpoint of the line segment \(PQ\). (1 mark) b) Use your result from part a) to determine the position vector representing the point \(S\) that lies two-thirds of the way along the line segment from \(M\) to \(R\). (2 marks)

The points \(P(2, -3, 5)\) and \(Q(8, 1, -3)\) lie in three-dimensional space. (a) Determine the position vector of the midpoint \(M\) of the line segment \(PQ\). (1 mark) (b) Use a vector method to verify that \(M\) divides \(PQ\) into two equal sections. (1 mark)

Points \(P\), \(Q\) and \(R\) have position vectors \(\vec{p} = 2\mathbf{i} - 3\mathbf{j} + 5\mathbf{k}\), \(\vec{q} = -4\mathbf{i} + 7\mathbf{j} + \mathbf{k}\) and \(\vec{r} = 8\mathbf{i} + \mathbf{j} - 3\mathbf{k}\) respectively. Point \(M\) is the midpoint of \(PQ\) and point \(N\) divides \(MR\) in the ratio \(2:3\). Determine the position vector of \(N\).