Parallel Vectors

# Parallel Vectors

GCSE(F), GCSE(H),

Normally, a vector ((a),(b)) is a position vector which describes a vector from the origin O to a point (a, b).

Points A, B and C are on the same line (they are said to be collinear). Vector vec(AB) = kvec(AC), where k is the scalar.

## Examples

1. Show that points A at (3, 5), B at (7, 11) and C at (13, 20) are collinear.

Answer: vec(AB) = ((4),(6)) and vec(AC) = ((10),( 15))

Let bb(a) = vec(AB) and bb(b) = vec(AC)

bb(a) = 2.5bb(b)

a is a multiple of b and both have the same origin; they are collinear.

2. Three points are on a straight line OB. Point O is at position (5, 7). Point B is at (-5, 12). Given vec(OB) = bb(b) and vec(OA) = bb(a), and bb(a) = frac(3)(5)bb(b), what is the position of point A?

vec(OB) = ((-10),(5))
vec(OA) = frac(3)(5) xx ((-10),(5)) = ((-6),(3))