These pages cover the longer vector questions in SPM Additional Mathematics, where a diagram gives ratios and you must find an unknown. The hard part is choosing the first step, not the algebra.
This hub sits inside the SPM Additional Mathematics vectors chapter. The basics in parallel vectors and collinearity and solving geometric vector ratios come first.
What is the one idea behind all of these?
Write the same vector in two different ways, then compare. Two routes to one point give two expressions, and because a and b are not parallel, their coefficients must match.
A short example shows it. Suppose M lies on AB, and OM is written once from the line AB and once as a multiple of a vector you were given. The first route gives (1 − λ)a + λb. The second gives, say, μ(a + 2b).
Matching the coefficients of a gives 1 − λ = μ. Matching b gives λ = 2μ. Then 1 − 2μ = μ, so μ = 1/3 and λ = 2/3. One line of reasoning produced both unknowns.
In what order should I study the lessons?
The four lessons build on each other, and each one uses a skill from the one before.
- Recovering an unknown ratio from two expressions for the same vector introduces equating coefficients.
- Distinguishing parallel from collinear statements in a diagram teaches what you may conclude from a diagram.
- Finding an internal intersection using two vector paths combines two routes for crossing lines.
- Checking the sign of an external division ratio handles points beyond the segment.
After the lessons, the integrated practice set mixes all four types.
Who should start where?
If you can do basic vector questions but diagram questions stall you at line one, start with lesson 1. If your working is right but the conclusion loses marks, go to lesson 2. If you can do intersections when the question gives a hint, try lesson 3 cold and compare.
If you want a teacher to watch your first step, see online one-to-one Additional Mathematics tuition. The mistake log and paper-error review tool shows which of the four topics costs you marks.