If a vector is written in two ways, the parts in the direction of a must match, and the parts in the direction of b must match. That gives two equations for two unknowns.
This lesson opens the vector dependence and ratio reasoning set. It needs the ratio idea from solving geometric vector ratios.
Why are the coefficients allowed to match?
Vectors a and b point in different directions, so neither is a multiple of the other. Then the only way xa + yb can equal zero is x = 0 and y = 0.
Moving everything to one side of an equation turns it into that form. That is why the step is valid, and also why it fails if a and b are parallel.
Worked example: a point on AB and on a line from O
The vectors are OA = a and OB = b, with a and b not parallel. M lies on AB, and OM = μ(a + 2b) for some number μ. Find the ratio AM:MB.
Route 1, along AB. Let AM = λAB. Then OM = a + λ(b − a) = (1 − λ)a + λb.
Route 2, given. OM = μa + 2μb.
Equate coefficients.
- Coefficient of a: 1 − λ = μ
- Coefficient of b: λ = 2μ
Substitute the second into the first: 1 − 2μ = μ, so μ = 1/3 and λ = 2/3.
AM = (2/3)AB, so MB = (1/3)AB, and AM:MB = 2:1.
Check: OM = (1/3)a + (2/3)b. The coefficients add up to 1, so M is on AB as required.
The mistake that loses the marks
A common slip is to stop at “λ = 2/3”. The question asks for a ratio.
| Step | Incomplete | Complete |
|---|---|---|
| Result | λ = 2/3 | λ = 2/3 |
| Interpretation | none | AM = (2/3)AB, MB = (1/3)AB |
| Answer | “2/3” | AM:MB = 2:1 |
Another slip is to equate a coefficient of a with a coefficient of b. Keep a and b in separate columns.
Check yourself
OA = a and OB = b, with a and b not parallel. M lies on AB and OM = μ(3a + b). Find AM:MB.
Answer
Route 1: OM = (1 − λ)a + λb. Route 2: OM = 3μa + μb.
Coefficient of a: 1 − λ = 3μ. Coefficient of b: λ = μ. So 1 − λ = 3λ, which gives λ = 1/4.
Then AM = (1/4)AB and MB = (3/4)AB, so AM:MB = 1:3.
Check: OM = (3/4)a + (1/4)b, and the coefficients add up to 1.
What to study next
Next, distinguishing parallel from collinear statements in a diagram shows what you may conclude from the information in a figure. The integrated practice set tests all four lessons.
If you want a teacher to watch you choose the two routes, see online one-to-one Additional Mathematics tuition. The word-problem structure worksheet can also help you lay the routes out before you start.