Theory question 115
Vectors(a) Given that m = (6i + 8j) and n = (−8i + (7/3)j), find the: (i) magnitudes and directions of m and n, (ii) angle between m and n.
(b) The position vectors of points P, Q, R, S are (−2, 3), (10, 4), (3, 12) and (4, 0) (column vectors) respectively. Show that PQ is pependicular to RS
Show the model answer
Model answer
(a)(i) |m| = 10, at tan⁻¹(8/6) = 53.13° to the positive x-axis (bearing 036.87°).
|n| = √(64 + 49/9) = 25/3 ≈ 8.33, at 180° − tan⁻¹(7/24) = 163.74° to the positive x-axis (bearing 286.26°).
(ii) m·n = −48 + 56/3 = −88/3; cos θ = (−88/3)/(10 × 25/3) = −0.352, θ ≈ 110.61°.
(b) PQ = (12, 1), RS = (1, −12); PQ·RS = 12 − 12 = 0, so PQ ⊥ RS.
Explanation
Two vectors are perpendicular when their dot product is zero.