Theory question 107
Coordinate GeometryThe coordinates of the vertices of triangle ABC are A(−2, 1), B(4, −2), and C(1, 8) respectively. If D(x, y) is the foot of the perpendicular from A to BC, find (A) an equation connecting x and y; (b) the unit vector in the direction of BC.
Show the model answer
Model answer
(a) Gradient of BC = (8 + 2)/(1 − 4) = −10/3
D lies on BC: y + 2 = −10/3(x − 4), i.e. 10x + 3y − 34 = 0
(Also AD ⟂ BC gives 3x − 10y + 16 = 0; together D = (292/109, 262/109))
(b) BC = −3i + 10j, |BC| = √109
Unit vector = (−3i + 10j)/√109
Explanation
D is on line BC, and AD is perpendicular to BC; divide a vector by its length to get a unit vector.