- A vector of magnitude OP in the direction from O to P is represented by OP. If OP−3OQ+2OR=0, show that P,Q,R are collinear.
Three points with position vectors a, b and c are collinear if and only if the vectors (a−b) and (a−c) are parallel.
In other words, to prove collinearity, we would need to show (a−b)=k(a−c) for some constant k.
For our example, we have OP−OQ=2(OQ−OR), and so OQ−OP=−2(OQ−OR), telling us that P,Q and R are collinear.
- A unit vector parallel to the x-axis is represented by i and a unit vector parallel to the y-axis by j. If OP=ai+sj and OQ=−ai+tj, 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 OQ when OP=2i+3j and OQ is perpendicular to OP.
The locus of P will be the line x=a, while the locus of Q will be x=−a. These are parallel straight lines.
The diagram shows the case a=2. The point P is at (2,3), and Q is at (−2,k).
We are told that OP and OQ are perpendicular, so the gradients of OP and OQ must multiply to −1.
We could alternatively use that OP.OQ=0.
Thus 32×k−2=−1⟹k=43. Thus OQ=−2i+43j.
