Theory question 116
Vectors(a) Find, correct to one decimal place, the angle between p = (3; −1) and q = (3; 4).
(b) ABCD is a square with vertices at A (0,0), B(2,0), C(2, 2) and D (0, 2). Forces of magnitudes 10N, 15N, 20N and 5N act along BA, BC, DC and AD respectively. Find the (i) magnitude; (ii) direction; of their resultant.
Show the model answer
Model answer
(a) p·q = 9 − 4 = 5; |p| = √10, |q| = 5
cos θ = 5/(5√10) = 0.3162, θ ≈ 71.6°
(b) BA: (−10, 0); BC: (0, 15); DC: (20, 0); AD: (0, 5)
Resultant = (10, 20)
(i) |R| = √(10² + 20²) = √500 ≈ 22.4 N
(ii) tan θ = 20/10 = 2, θ ≈ 63.4° to AB (the positive x-direction), i.e. a bearing of about 026.6°
Explanation
Write each force as a vector using the direction of the side it acts along, then add.