1. A vector of magnitude \(OP\) in the direction from \(O\) to \(P\) is represented by \(\mathbf{OP}\). If \(\mathbf{OP}-3\mathbf{OQ}+2\mathbf{OR}=\mathbf{0}\), show that \(P\),\(Q\),\(R\) are collinear.

If three points \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are collinear, can we find a connection between \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\)?

  1. A unit vector parallel to the \(x\)-axis is represented by \(\mathbf{i}\) and a unit vector parallel to the \(y\)-axis by \(\mathbf{j}\). If \(\mathbf{OP}=a\mathbf{i}+s\mathbf{j}\) and \(\mathbf{OQ}=-a\mathbf{i}+t\mathbf{j}\), where \(a\) is a constant and \(s\) and \(t\) are variables, show that the loci of \(P\) and \(Q\) are parallel straight lines. In this case find \(\mathbf{OQ}\) when \(\mathbf{OP}=2\mathbf{i}+3\mathbf{j}\) and \(OQ\) is perpendicular to \(OP\).

Can we draw a helpful diagram? What is the gradient of each line? What do we know if two lines are perpendicular?