A position vector describes a point's location relative to a fixed origin O. Writing OA as a column vector from O to A gives the standard GCSE notation. Position vectors underpin vector path problems — finding midpoints, expressing journeys between points, and proving lines are parallel or points are collinear.
What is a position vector?
A position vector gives the displacement from the origin O to a specific point. It is always measured from O (the origin, with coordinates (0, 0)) to the point in question.
For a point A with coordinates (a₁, a₂), the position vector is:
OA⃗ = (a₁ choose a₂) (column vector notation)
The vector OA⃗ is often written in bold as a (in textbooks) or underlined as a (in handwriting). The arrow above OA indicates direction — from O to A.
Key distinction: A position vector starts at the origin. A general vector can start anywhere and describes a displacement.
How do you find the vector from one point to another?
To find the vector from point A to point B:
AB⃗ = OB⃗ − OA⃗ = b − a
This follows from the vector path: O → A → B is the same as going backwards along OA then forwards along OB.
Worked example 1
A has position vector (2 choose 5) and B has position vector (8 choose 1).
AB⃗ = OB⃗ − OA⃗ = (8 choose 1) − (2 choose 5) = (6 choose −4)
Check: starting at A(2, 5), moving 6 right and 4 down lands at (8, 1) = B ✓
How do you find the midpoint of two points using position vectors?
The position vector of the midpoint M of AB is the average of the two position vectors:
OM⃗ = ½(OA⃗ + OB⃗) = ½(a + b)
Worked example 2
A has position vector (4 choose 6) and B has position vector (10 choose 2).
OM⃗ = ½ [(4 choose 6) + (10 choose 2)] = ½ (14 choose 8) = (7 choose 4)
The midpoint M has coordinates (7, 4).
This matches the coordinate midpoint formula: ((4+10)/2, (6+2)/2) = (7, 4) ✓ — the vector and coordinate approaches agree.
How are position vectors used in vector path problems?
Vector path problems at GCSE Higher give you several position vectors and ask you to express a journey in terms of them. Use the principle that any path from A to B can be rerouted through O:
AB⃗ = −OA⃗ + OB⃗ = OB⃗ − OA⃗ = b − a
Worked example 3
OA⃗ = a, OB⃗ = b. Point M divides AB in the ratio 2:1.
First find AB⃗ = b − a.
Since M is ⅔ of the way from A to B:
AM⃗ = ⅔ × AB⃗ = ⅔(b − a).
OM⃗ = OA⃗ + AM⃗ = a + ⅔(b − a) = a + ⅔b − ⅔a = ⅓a + ⅔b
| Path | Vector expression |
|---|---|
| OA⃗ | a |
| OB⃗ | b |
| AB⃗ | b − a |
| BA⃗ | a − b |
| Midpoint M of AB | ½a + ½b |
| Point dividing AB in ratio m:n from A | a + m/(m+n) × (b − a) |
What is the magnitude of a position vector?
The magnitude (length) of position vector OA⃗ = (a₁ choose a₂) is:
|OA| = √(a₁² + a₂²)
This is just the distance from the origin to the point, by Pythagoras. For example, OA = (3 choose 4) gives |OA| = √(9 + 16) = √25 = 5.
Frequently asked questions
What is the difference between OA⃗ and AO⃗?
OA⃗ goes from O to A — it is the position vector of A. AO⃗ goes from A to O — it is the negative of the position vector: AO⃗ = −OA⃗. Direction matters in vectors, so always read the subscript letters in order to determine the direction of travel.
Do I need to know both column vector notation and letter notation?
Yes. GCSE questions use column vectors like (3 choose −2) for specific calculations, and letter notation like a and b for general proofs and expressions. Make sure you can work fluently in both. Letter notation is especially common in "show that two vectors are parallel" or "show that points are collinear" questions.
How do you show that three points are collinear using position vectors?
Show that one vector (say AB⃗) is a scalar multiple of another (say BC⃗). If AB⃗ = k × BC⃗ for some scalar k, then AB and BC are parallel and share point B — so A, B, C lie on the same straight line (are collinear). State the scalar explicitly and conclude "since AB⃗ is parallel to BC⃗ and they share point B, A, B, C are collinear."
Are position vectors on the foundation or higher tier?
Vectors are a Higher tier topic at GCSE. Position vectors specifically — along with ratio division of a line and proving collinearity — are assessed at Higher tier. Foundation students are not expected to use vector notation, though they use the underlying coordinate ideas throughout their course.
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