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) Find the ratio of (|overrightarrow{PQ}| : |overrightarrow{QR}|)
Explanation
(a) For P, Q and R to lie on a straight line,the gradient of any two of PQ, PR and QR must be equal.
Gradient of PR : (frac{6 – 1}{2 – 11} = -frac{5}{9})
Gradient of QR : (frac{6 – frac{13}{3}}{2 – 5} = frac{frac{5}{3}}{-3})
= (- frac{5}{9})
The gradients are the same therefore P, Q and R are on the same line.
(b) (overrightarrow{PQ} = (5i + frac{13}{3}j) – (11i + j))
= (-6i + frac{10}{3}j)
(|overrightarrow{PQ}| = sqrt{(-6)^{2} + (frac{10}{3})^{2}})
= (sqrt{36 + frac{100}{9}})
= (sqrt{frac{424}{9}})
(overrightarrow{QR} = (2i + 6j) – (5i + frac{13}{3}j))
= (-3i + frac{5}{3}j)
(|overrightarrow{QR}| = sqrt{(-3)^{2} + (frac{5}{3})^{2}})
= (sqrt{9 + frac{25}{9}})
= (sqrt{frac{106}{9}})
(frac{|overrightarrow{PQ}|}{|overrightarrow{QR}|} = frac{sqrt{frac{424}{9}}}{sqrt{frac{106}{9}}})
= (frac{4}{1})
(therefore |overrightarrow{PQ}| : |overrightarrow{QR}| = 4 : 1)