The position vectors of points P, Q and R with respect to the origin are ((4i – 5j), (i + 3j)) and ((-5i + 2j)) respectively. If PQRM is a parallelogram, find:
(a) the position vector of M ;
(b) (|overrightarrow{PM}|) and (|overrightarrow{PQ}|) ;
(c) the acute angle between (overrightarrow{PM}) and (overrightarrow{PQ}), correct to 1 decimal place ;
(d) the area of PQRM.
Explanation
Since PQRM is a parallelogram,
(overrightarrow{PQ} = overrightarrow{MR})
((4 – 1)i + (-5 – 3)j = (x + 5)i + (y – 2)j)
Equating components, we have
(3 = x + 5 implies x = 3 – 5 = -2)
(-8 = y – 2 implies y = -8 + 2 = -6)
(therefore M = (-2i – 6j))
(b) (|overrightarrow{PM}| = sqrt{(-2 – 4)^{2} + (- 6 + 5)^{2}} = sqrt{37})
(|overrightarrow{PQ}| = sqrt{(1 – 4)^{2} + (3 + 5)^{2}} = sqrt{73})
(c) Let the angle be (theta).
(overrightarrow{PM} . overrightarrow{PQ} = |PM| |PQ| cos theta)
((-6i – j) . (-3i + 8j) = (sqrt{37})(sqrt{73}) cos theta)
(18 – 8 = sqrt{2701} cos theta)
(cos theta = frac{10}{sqrt{2701}} = 0.1924)
(theta = 78.907° approxeq 78.9°) (to 1 d.p)
(d) Area of PQRM = ((sqrt{37})(sqrt{73}) sin 78.907°)
= (sqrt{2701} sin 78.907°)
= (51 text{sq. units})