Theory question 117
Vectors(a) given that P = (4i − 3j) and q = (−i + 5j), find r such that |r| = 15 and is in the direction of (2p + 3q)
(b) Forces of magnitude 8N, 6N and 4N act at the point P, as shown in the diagram. Find the: (i) magnitude; (ii) direction of the resultant force.
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(a) 2p + 3q = (8i − 6j) + (−3i + 15j) = 5i + 9j, |5i + 9j| = √106
r = 15(5i + 9j)/√106 ≈ 7.28i + 13.11j
(b) Take i to the right and j along the 6N force; 8N is 30° left of it and 4N is 60° right of it
Σx = −8 sin 30° + 4 sin 60° = −4 + 2√3 ≈ −0.536
Σy = 6 + 8 cos 30° + 4 cos 60° = 8 + 4√3 ≈ 14.928
(i) R = √(0.5362 + 14.9282) ≈ 14.9 N
(ii) tan α = 0.536/14.928, α ≈ 2.1° to the left of the 6N force (about 92.1° to the horizontal line through P, measured from the right)
Resolve each force along and across the 6N direction.