Theory question 114
Vectors(a) Five forces acting on a particle are represented by (2i + 3j), (4i − 7j), (−5i + 8j), (i + j) and (pi + qj), where p, q ∈ R. If the particle is in equilibrium, find the values of p and q. (b) WXYZ is a parallelogram whose vertices are W(−1, 1), X(m, n), Y(5, −7) and Z(2, 4). Find the: (i) values of m and n; (ii) unit vector in the direction of XY.
Show the model answer
Model answer
(a) Sum of forces = 0: i: 2 + 4 − 5 + 1 + p = 0, p = −2; j: 3 − 7 + 8 + 1 + q = 0, q = −5.
(b)(i) Diagonals of a parallelogram bisect each other: W + Y = X + Z, so (4, −6) = (m + 2, n + 4); m = 2, n = −10.
(ii) XY = (5 − 2)i + (−7 + 10)j = 3i + 3j; unit vector = (3i + 3j)/(3√2) = (i + j)/√2.
Explanation
In equilibrium the i and j parts each add up to zero.