Position Vectors

A position vector tells us where a point is, measured from the origin O.

The origin is the point (0, 0). If a point is called P, then the position vector of P is the vector from O to P. It is written as OP.

Position vectors connect coordinates with column vectors.

What you will learn

• Understand what a position vector is
• Write the position vector of a point
• Find coordinates from a position vector
• Find the vector joining two points
• Use the rule finish minus start
• Use correct vector notation with an arrow above the letters

Key Ideas

Position Vector of a Point

If a point has coordinates (x, y), its position vector is written as a column vector with x on top and y below.

Point	Position Vector
P(a, b)	OP =
Q(−3, 6)	OQ =

Diagram: Position Vector

The diagram below shows the position vector of P(5, −2). The vector starts at the origin O and ends at point P.

The vector OP starts at O and finishes at P.

Worked Example 1: Writing a Position Vector

Example 1: Write the position vector of P(5, −2).

1. The point is P(5, −2).
2. The x-coordinate is 5.
3. The y-coordinate is −2.
4. Write the x-coordinate on top and the y-coordinate below.

Answer:

OP =

From Position Vector to Coordinates

You can also work backwards. If you are given the position vector, you can find the coordinates of the point.

Example 2: The position vector of P is OP = . State the coordinates of P.

1. The top entry is the x-coordinate.
2. The bottom entry is the y-coordinate.
3. So x = 2 and y = 7.

Answer: P(2, 7)

Finding the Vector Joining Two Points

To find the vector from one point to another, subtract the position vectors.

The vector AB starts at A and finishes at B. So B is the finish and A is the start.

Vector	Start	Finish	Rule
AB	A	B	OB − OA
PQ	P	Q	OQ − OP

Diagram: Finish Minus Start

The diagram below separates the three vectors clearly:

• OA goes from the origin to A.
• OB goes from the origin to B.
• AB goes from A to B.

Worked Example 3: Finding AB

Example 3: A is (1, 3) and B is (4, 7). Find AB.

1. Write the position vector of A.

OA =

2. Write the position vector of B.

OB =

3. Use finish minus start.

AB = OB − OA

AB = −

4. Subtract matching entries.

• Top entry: 4 − 1 = 3
• Bottom entry: 7 − 3 = 4

Answer:

AB =

Worked Example 4: Finding PQ

Example 4: P is (0, 4) and Q is (3, −1). Find PQ.

Use:

OP = , OQ =

PQ = −

• Top entry: 3 − 0 = 3
• Bottom entry: −1 − 4 = −5

Answer:

PQ =

Worked Example 5: Subtracting a Negative Number

Example 5: E is (−3, 2) and F is (1, −4). Find EF.

Use finish minus start: EF = OF − OE

OE = , OF =

EF = −

• Top entry: 1 − (−3) = 1 + 3 = 4
• Bottom entry: −4 − 2 = −6

Answer:

EF =

Common Mistakes

• Writing the y-coordinate on top instead of the x-coordinate.
• Forgetting that a position vector starts from the origin.
• Using start minus finish instead of finish minus start.
• Mixing up AB and BA.
• Forgetting to subtract both the top and bottom entries.
• Making sign errors when subtracting negative numbers.

Exam-style multiple choice

Practice Questions

1. Write the position vector of the point R(4, −8).
2. A is (2, 1) and B is (7, 5). Find AB.
3. S is (2, −3) and T is (−1, 1). Find ST.
4. The position vector of M is OM = . State the coordinates of M.
5. C is (−2, 5) and D is (4, 1). Find CD.

Answers

1. OR =
2. AB =
3. ST =
4. M(−6, 3)
5. CD =