Theory question 116
Vectors(a) The position vectors of points L and M are (5i + 6j) and (13i + 4j) respectively. If point K lies on LM such that LK : KM is 2:3, find the position vector of K.
(b) Three poles are situated at points A, B and C on the same horizontal plane such that AB = (8km, 060°) and BC = (12km, 130°). Calculate: (i) |AB|, correct to three decimal places (ii) the bearing of C from A, correct to the nearest degree.
Show the model answer
(a) LM = 8i − 2j; LK = (2/5)LM = (16/5)i − (4/5)j
OK = 5i + 6j + (16/5)i − (4/5)j = (41/5)i + (26/5)j
(b) Angle ABC = 180° − 130° + 60° = 110°
(i) |AC|2 = 82 + 122 − 2(8)(12)cos 110° = 273.67, |AC| = 16.543 km (|AB| itself is the given 8 km)
(ii) AC = (8 sin 60° + 12 sin 130°)i + (8 cos 60° + 12 cos 130°)j = 16.121i − 3.714j
Bearing of C from A = 90° + tan−1(3.714/16.121) ≈ 103°
Use the cosine rule in triangle ABC, then resolve the displacements east and north to get the bearing.