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?

Answer: (-1, 10)

`vec(OB) = ((-10),(5)) `

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

Origin + vector a = (5 - 6, 7 + 3) = (-1, 10)