Theory question 114
VectorsThe position vectors of 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 coordinates of M; (b) the acute angle between PM and PQ, correct to the nearest degree.
Show the model answer
Model answer
(a) In PQRM, PM = QR, so M = P + R − Q = (4 − 5 − 1, −5 + 2 − 3) = (−2, −6).
(b) PM = −6i − j, PQ = −3i + 8j. Dot product = 18 − 8 = 10.
cos θ = 10/(√37 × √73) ≈ 0.1924, θ ≈ 79°.
Explanation
Opposite sides of a parallelogram are equal vectors.