(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) |AC|, correct to three significant figures ; (ii) the bearing of C from A, correct to the nearest degree.

The coplanar forces 4N, 8N, 6N, 4N, and 5N act at a point as shown in the diagram. If the 6N force acts in the direction 090°, calculate the :

(a) magnitude of the resultant force;

(b) direction of the resultant force.

(a) The position vectors of points A, B and C are \(i + 5j , 3i + 9j\) and \(-i + j\) respectively. (i) Show that points A, B and C are collinear; (ii) Determine the ratio \(|AB| : |BC|\).

(b) A uniform beam XY of mass 10 kg and length 24m is hunged horizontally from a cross bar by teo vertical inextensible strings, one attached to X and the other at a point M, 4m away from Y. A mass of 50kg is suspended at a point N which is 8m from X. If the system remains in equilibrium, calculate the tensions in the strings.