Position Vectors, Displacement and Distance — Edexcel A-level Mathematics (Pure)

SPEC 10.4 • VECTORS

Position Vectors and Displacement

Locating points and finding distances using vectors

What is a Position Vector?

A position vector is the vector from the origin O to a point.

Position Vector

The position vector of point P is the vector

OP ⟶

from the origin O to P.

We write it as p (lowercase bold) or

OP ⟶

Position Vector Notation

If a point has coordinates, its position vector has the same components:

2D Position Vectors

Point A(3, 5) has position vector:

a = (35) = 3 i + 5 j

3D Position Vectors

Point B(2, −1, 4) has position vector:

b = 2 - 1 4 = 2 i - j + 4 k

Convention

We use lowercase bold letters for position vectors: a, b, p

The position vector of point A is written as a (or

O A ⟶

)

Displacement Vectors

A displacement vector goes from one point to another (not necessarily from the origin).

Key Formula

A B ⟶ = b - a

The displacement from A to B = (position vector of B) − (position vector of A)

Order Matters!

$A B ⟶ = b - a$ — "destination minus start"

$B A ⟶ = a - b$ — going the other way gives the negative

Note:

A B ⟶ = - B A ⟶

Why Does This Work?

To travel from A to B, we can go via the origin O:

Derivation

Starting at A:

Go from A back to O: this is −a (reverse of position vector a)

Then go from O to B: this is +b (position vector b)

Total journey:

A B ⟶ = - a + b = b - a

✓

Worked Example01

Write the position vector of P(3, 5) in column form and i-j notation.

1

The position vector of P is the vector from O to P

2

In column form:

p = (35)

3

In i-j notation:

p = 3 i + 5 j

Answer

p = (35) = 3 i + 5 j

Worked Example02

Given A(2, 1) and B(5, 4), find the displacement vector $A B ⟶$ .

1

Write position vectors:

a = (21)

,

b = (54)

2

Apply formula:

A B ⟶ = b - a

3

Calculate:

A B ⟶ = (54) - (21) = (5 - 2 4 - 1) = (33)

Answer

A B ⟶ = (33) = 3 i + 3 j

Distance Between Two Points

The distance between two points is the magnitude of the displacement vector.

Distance Formula (2D)

A B ⟶ = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Method

Step 1: Find

A B ⟶ = b - a

Step 2: Find

A B ⟶ = x^{2} + y^{2}

where the components are x and y

Alternative View

This is the same as Pythagoras in the coordinate plane!

Worked Example03

Find the distance between A(1, 3) and B(4, 7).

1

Find displacement:

A B ⟶ = (4 - 1 7 - 3) = (34)

2

Find magnitude:

A B ⟶ = 3^{2} + 4^{2}

3

Calculate:

A B ⟶ = 9 + 16 = 25 = 5

AnswerDistance = 5 units

Distance in 3D

The same method extends naturally to three dimensions:

Distance Formula (3D)

A B ⟶ = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}

Same Process

Step 1: Find

A B ⟶ = b - a

(subtract each component)

Step 2: Find

A B ⟶ = x^{2} + y^{2} + z^{2}

Remember

It's just Pythagoras extended to 3D — sum of squares, then square root!

Worked Example04

Find $PQ ⟶$ and $PQ ⟶$ for P(1, 2, 3) and Q(4, 6, 3).

1

Find displacement:

PQ ⟶ = q - p = 4 - 1 6 - 2 3 - 3 = 340

2

Find magnitude:

PQ ⟶ = 3^{2} + 4^{2} + 0^{2}

3

Calculate:

PQ ⟶ = 9 + 16 + 0 = 25 = 5

Answer

PQ ⟶ = 340 = 3 i + 4 j

, Distance = 5 units

Midpoint Using Position Vectors

The midpoint of two points can be found using their position vectors:

Midpoint Formula

m = \frac{1}{2} (a + b)

The position vector of M is the average of position vectors of A and B

In Coordinates

Midpoint of A(x₁, y₁) and B(x₂, y₂):

M =

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Simple Idea

Just average each coordinate — halfway between each pair of values!

Worked Example05

Find the midpoint of A(2, 6) and B(8, 2) using position vectors.

1

Write position vectors:

a = (26)

,

b = (82)

2

Add vectors:

a + b = (2 + 8 6 + 2) = (108)

3

Find midpoint:

m = \frac{1}{2} (108) = (54)

AnswerMidpoint M = (5, 4)

Worked Example06

M(4, 5) is the midpoint of A(1, 3) and B. Find the coordinates of B.

1

Use the midpoint formula:

m = \frac{1}{2} (a + b)

2

Rearrange:

2 m = a + b

, so

b = 2 m - a

3

Substitute:

b = 2 (45) - (13) = (810) - (13) = (77)

AnswerB = (7, 7)

Practice

Question 1: Given A(3, 1) and B(7, 4), find:

(a) the displacement vector

A B ⟶

(b) the distance AB

(a) Displacement:

A B ⟶ = (7 - 3 4 - 1) = (43)

(b) Distance:

A B ⟶ = 4^{2} + 3^{2} = 16 + 9 = 25 = 5

units

Practice

Question 2: Find the midpoint of P(2, 4, 6) and Q(8, 2, 10) using position vectors.

Method:

m = \frac{1}{2} (p + q)

p + q = 246 + 8210 = 10616

m = \frac{1}{2} 10616 = 538

AnswerMidpoint M = (5, 3, 8)

Practice

Question 3: M(5, 2) is the midpoint of A(2, −1) and B. Find the coordinates of B.

Using:

b = 2 m - a

b = 2 (52) - (2 - 1)

b = (104) - (2 - 1) = (10 - 2 4 - (- 1)) = (85)

AnswerB = (8, 5)

Summary

Key Formulas

ConceptFormula

Position Vector of P(x, y)

p = (x y)

Displacement A to B

A B ⟶ = b - a

Distance (2D)

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Distance (3D)

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}

Midpoint

m = \frac{1}{2} (a + b)

Key Exam Skills

Order matters:

A B ⟶ = b - a

(destination − start)

Distance = magnitude of displacement vector

Reverse midpoint:

b = 2 m - a