The position vectors of P, Q and R are \(11i + j, 5i + \frac{13}{3}j\) and \(2i + 6j\) respectively. (a) Show that P, Q and R lie on a straight line. (b) Fi...

Answer Details

The position vectors give \(P(11,1)\), \(Q\left(5,\tfrac{13}{3}\right)\), \(R(2,6)\).

(a) Show P, Q, R are collinear

Find the displacement vectors \(\overrightarrow{PQ}\) and \(\overrightarrow{QR}\):

\[\overrightarrow{PQ}=Q-P=\left(5-11,\;\tfrac{13}{3}-1\right)=\left(-6,\;\tfrac{10}{3}\right),\]

\[\overrightarrow{QR}=R-Q=\left(2-5,\;6-\tfrac{13}{3}\right)=\left(-3,\;\tfrac{5}{3}\right).\]

Compare them: \(\overrightarrow{PQ}=2\left(-3,\tfrac{5}{3}\right)=2\,\overrightarrow{QR}\). Since \(\overrightarrow{PQ}\) is a scalar multiple of \(\overrightarrow{QR}\), the two vectors are parallel; and because they share the common point \(Q\), the points \(P\), \(Q\) and \(R\) lie on one straight line.

(b) Ratio \(|\overrightarrow{PQ}|:|\overrightarrow{QR}|\)

From \(\overrightarrow{PQ}=2\,\overrightarrow{QR}\), the magnitudes satisfy \(|\overrightarrow{PQ}|=2\,|\overrightarrow{QR}|\). Hence

\[|\overrightarrow{PQ}|:|\overrightarrow{QR}|=2:1.\]