Every vector proof works the same way: express both vectors in terms of the two base vectors, then compare them. If PQ = k × RS for some number k, the two lines are parallel, and if they also share a point they are collinear. For a point P that divides AB in the ratio m : n, the position vector is OP = a + [m/(m+n)](b - a). The algebra is short. The marks are in stating the geometric conclusion afterwards.
Students often do all the algebra correctly and then score badly, because they stop at the algebra. A vector proof is a geometric argument, and the final line has to say what has been proved: that the lines are parallel, that the points are collinear, that the shape is a parallelogram. Leaving the examiner to infer it costs the concluding mark.
The other difficulty is starting. Faced with a diagram covered in letters, students do not know which vector to write first. The answer is always the same: express everything in terms of the two vectors you were given, and do not introduce anything else.
Step 1
express every vector using a and b only
Step 2
compare the two vectors you care about
Parallel
PQ = k × RS for some number k
Collinear
parallel and sharing a common point
Ratio point
OP = a + [m/(m+n)](b - a)
Step 3
state the conclusion in words
The route rule AB = b - a does most of the work. If you can express any vector in the diagram as a chain of journeys through known points, the rest is collecting like terms, exactly as in simplifying algebraic expressions.
In a quadrilateral PQRS, PQ = 3a + 2b and SR = 3a + 2b. Prove that PQRS is a parallelogram.
Solution
PQ = 3a + 2b and SR = 3a + 2b.PQ = SR.PQ is parallel to SR, and PQ is the same length as SR.Equal versus parallel: if two vectors are equal, the lines are parallel and the same length. If one is a scalar multiple of the other, such as PQ = 2 × SR, they are parallel but different lengths. Read carefully which one the question needs, because a parallelogram proof requires equality, not just a multiple.
OA = a and OB = b. The point P lies on AB such that AP : PB = 2 : 3. Express OP in terms of a and b.
Solution
2 : 3 splits AB into 2 + 3 = 5 equal parts.2/5 of the way from A to B.OP = OA + AP.AB = b - a, so AP = ⅖(b - a).OP = a + ⅖(b - a).OP = a + ⅖b - ⅖a = ⅗a + ⅖b, that is OP = (3/5)a + (2/5)b.The step students get wrong
Using the ratio numbers directly as fractions. In the ratio 2 : 3 the point is 2/5 of the way along, not 2/3. Always add the two parts of the ratio together first to find the total number of parts, then form the fraction. Writing 2 + 3 = 5 on the page before anything else prevents this.
The coefficients in a correct ratio-point answer always add up to 1. In the example above, 3/5 + 2/5 = 1. That is not a coincidence: any point lying on the line AB has this property, so it is a fast way to catch an arithmetic slip.
The midpoint case is worth memorising on its own, since it appears constantly: the position vector of the midpoint of AB is ½(a + b), the average of the two endpoints, matching the midpoint formula in coordinate geometry.
OA = a, OB = b. M is the midpoint of OA and N is the midpoint of OB. Prove that MN is parallel to AB and half its length.
Solution
M is the midpoint of OA, so OM = ½a.N is the midpoint of OB, so ON = ½b.MN = ON - OM = ½b - ½a = ½(b - a).AB = b - a.MN = ½ × AB, a scalar multiple.That result is the midpoint theorem, proved in four lines of vector algebra. The same comparison, one vector as a scalar multiple of another, is what proves points lie on a straight line in position vectors and collinearity.
What exactly proves that two lines are parallel?
Showing one vector is a scalar multiple of the other, so PQ = k × RS where k is any number. The value of k also tells you the length ratio: k = ½ means half as long.
What is the difference between parallel and collinear?
Parallel lines point the same way but can be anywhere on the page. Collinear points lie on one single straight line, which needs the vectors to be parallel and to share a common point. Always name the shared point in a collinearity proof.
How do I turn a ratio into a fraction?
Add the parts to get the total, then the first part over that total is how far along you are. For 3 : 1 the total is 4, so the point is 3/4 of the way from the first named point to the second.
Do I have to write a conclusion in words?
Yes, and it usually carries its own mark. A proof that ends with an equation has shown a fact but has not stated what it means. One sentence naming the geometric result is enough.
How does this connect to the rest of E-Maths?
Vector proofs are congruence and similarity arguments done with algebra instead of angle-chasing, so they overlap with similarity tests. The routes themselves are built from adding and subtracting vectors.
— Mr Gan Math Tuition
Mr. Gan teaches the two-step structure so every proof follows the same shape.
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