Theory question 117
Vectors(a) m(2 / 1) + n(−1 / 2) = (5 / −1) where m and n are scalars. Find the value of (m + n). (Column vectors are written top / bottom.)
(b) A(−1, 3), B(2, −1) and C(5, 3) are the vertices of ABC. (i) Express in column notation, the unit vectors parallel to AB and AC. (ii) Use a dot product to calculate BÂC, correct to the nearest degree.
Show the model answer
Model answer
(a) 2m − n = 5 and m + 2n = −1
Double the first and add: 5m = 9, m = 9/5; n = 2m − 5 = −7/5
m + n = 2/5
(b)(i) AB = (3 / −4), |AB| = 5, unit vector (3/5 / −4/5)
AC = (6 / 0), |AC| = 6, unit vector (1 / 0)
(ii) cos BÂC = (AB · AC)/(|AB||AC|) = 18/30 = 0.6
BÂC ≈ 53°
Explanation
Use AB = B − A and AC = C − A, not the position vectors themselves.