A vector proof shows a geometric fact, such as two lines being parallel or three points lying on one line, by writing every journey between points as a combination of two base vectors. It is algebra with a diagram behind it.
The topic sits in the vectors part of Additional Mathematics. Check the current 0606 specification for the examination-year wording and given formulae.
What do you need before starting?
You should already be comfortable with these ideas:
- adding and subtracting vectors, and multiplying a vector by a number
- column vectors such as (3, −2) and their magnitude
- fractions and simple simultaneous equations, because intersection problems need both
If fractions slow you down, revise them first. Exact arithmetic practice with the non-calculator working trainer helps.
One orienting example
In triangle OAB, →OA = a and →OB = b. M is the midpoint of AB. Find →OM.
The journey from O to M can go O to A, then A to M. The vector →AB is b − a, so →AM is half of that.
→OM = →OA + →AM = a + ½(b − a) = ½a + ½b
That is the whole method in small: choose a route from a known start, write each leg using a and b, then simplify. Every lesson below builds on it.
In what order should you study the lessons?
- Express a displacement using base vectors: the route-writing habit that all later proofs depend on.
- Find a dividing point on a segment: uses a ratio to place a point on a line, and gives the position vector.
- Show points are collinear using a scalar relation: turns “on one line” into a multiple and a shared point.
- Find an intersection of vector-defined lines: compares coefficients of a and b with two different parameters.
- Explain direction in a vector equation: separates the position vector from the direction vector, and tests parallel lines.
Lessons 1 and 2 are needed for everything else. Lesson 5 can be done earlier if vector equations of lines feel unfamiliar.
What are the common traps?
- Reversed direction. →AB is b − a, not a − b. Always write “end minus start”.
- Weights on the wrong vector. On a ratio question, the point nearer A has more of a.
- Collinear without a common point. A multiple only shows parallel. You also need a shared point.
- One parameter for two lines. Two lines need two letters, say λ and μ.
- No conclusion sentence. A proof ends by saying what has been shown.
How should you use the practice set?
Work through the two-dimensional vector proofs practice set on paper, with a sketch for each question. Cover the answer, finish your own proof, then compare. Keep a note of each slip with the mistake log and retest queue so you can retest the same skill a few days later.
If you want a teacher to look at how you plan a proof, our online one-to-one Additional Mathematics tuition starts with a paid one-hour trial from RM80.