Theory question 114
VectorsThe position vectors of points A, B and C with respect to the origin are (8i − 10j), (2i + 6j) and (−10i + 4j) respectively. If ABCN is a parallelogram, find (a) the position vector of N. (b) |AN| and |AB|. (c) correct to two decimal places, the acute angle between AN and AB.
Show the model answer
Model answer
(a) In ABCN, AN = BC, so N = A + C − B = (8 − 10 − 2)i + (−10 + 4 − 6)j = −4i − 12j.
(b) AN = −12i − 2j, |AN| = √148 ≈ 12.17; AB = −6i + 16j, |AB| = √292 ≈ 17.09.
(c) AN·AB = 72 − 32 = 40; cos θ = 40/(√148 × √292) ≈ 0.1924, θ ≈ 78.91°.
Explanation
Opposite sides of a parallelogram are equal vectors.