Theory question 116
VectorsThe 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) |PM| and |PQ|; (c) the acute angle between PM and PQ, correct to one decimal place; (d) the area of PQRM.
Show the model answer
Model answer
(a) OM = OP + OR − OQ = −2i − 6j
(b) PM = −6i − j, |PM| = √37; PQ = −3i + 8j, |PQ| = √73
(c) PM·PQ = 18 − 8 = 10; cos θ = 10/(√37 √73) = 0.1924; θ = 78.9°
(d) Area = |PM||PQ| sin θ = √37 × √73 × sin 78.9° = 51 square units
Explanation
In parallelogram PQRM, PM = QR, so M = P + R − Q.