Question

This is an example in the text book.

Using vectors, demonstrate that the three points A(5, -1), B(-3,4) and C(13,-6) are collinear.

Solution
AB = (-8, 5)
BC = (16, -10)
Then BC = 2AB

AB and BC have the opposite direction, so the points A, B, and C must be collinear.

I don't understand how AB = (-8, 5) and
BC = (16, -10)

Answer 1

vector AB is (-3 -5) i + (4 - (-1) ) j
= -8 i + 5 j

vector BC is (13 - (-3)) i + (-6 - 4) j
= 16 i - 10 j

Answer 2

Now the lines are collinear because they go through the same point (point B)
and they have the same slope (although one is going down the hill while the other goes up)

Answer 3

I'm not sur if this will help, but to get AB, you subtract A from B and B from C.

B(-3x,4y)-A(5x,-1y)=x(-3-5),y(4+1)=
AB(-8,5)

C(13x,-6y)-B(-3x,4y)=x(13+3),y(-6-4)=
BC(16,-10)