Using multiples to show lines are parallel or points lie on a straight line, and matching amounts of a and b.
What you will learn
- Show that two lines are parallel
- Show that three points lie on a straight line
- Find an unknown by matching amounts of a and b
Parallel means a multiple
If PQ = 2a − 4b and RS = −a + 2b, then PQ = −2 × RS.
A multiple means parallel. Say so in words: "PQ is a multiple of RS, so PQ is parallel to RS."
To test, find the multiplier for the a parts and the multiplier for the b parts: they must be the same number. For 3a − 6b and a − 2b: 3 ÷ 1 = 3 and −6 ÷ (−2) = 3, so they are parallel. For a + kb to be parallel to 4a + 6b, the multiplier is 1/4 (from the a parts), so k = 1/4 × 6 = 3/2.
To show two lines in a shape are parallel, write each one in terms of a and b by a route, simplify, and compare. In a parallelogram OABC with P on OC and Q on AB, both one third of the way up, PQ = PO + OA + AQ = −(1/3)b + a + (1/3)b = a: PQ is parallel to OA.
The rest of this lesson and its 8 practice questions are locked.
In this lesson
- Parallel means a multiple
- On a straight line
- Matching a and b
- Try it: First question
8 practice questions, marked as you go; about 25 minutes in all.
In this unit: Vectors
Column vectors and how to add, subtract and multiply them, vectors between points, and vector geometry and proof in terms of a and b.