11.4PureCore

Position Vectors

A position vector pins a point to the origin. Once points have position vectors, the vector joining any two of them follows from one simple rule — the key to all the geometry problems in the next lesson.

25 min Video by Zeeshan Zamurred Vectors

Edexcel AS Level Maths: 11.4 Position VectorsWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand what a position vector is
Write the position vector of a point
Find the vector between two points
Find the distance between two points using vectors

Position vectors

The $position vector$ of a point $A$ is $O A$ , the vector from the origin $O$ to $A$ . It is usually written $a$ . So the point $(3, 5)$ has position vector $(35)$ .

The vector joining two points

To get from $A$ to $B$ , go back to the origin then out to $B$ : $A B = b - a$ . Always “end minus start”.

A B = b - a

End point’s position vector minus the start point’s.

A B = (47) - (12)

Answer

(35)

Tip — AB = b − a, not a − b. Getting the order wrong reverses the vector.

Distance between points

The distance from $A$ to $B$ is the magnitude of $A B$ — exactly the distance formula in disguise.

∣ A B ∣ = ∣ b - a ∣

Find the joining vector, then its magnitude.

A B = (34)

2Magnitude

= 9 + 16 = 5

Answer

5

Formula recap

O A = a

Position vector of A.

A B = b - a

Vector joining two points.

∣ A B ∣ = ∣ b - a ∣

Distance between points.

Common mistakes to avoid

Writing AB = a − b.

It is end minus start: AB = b − a.

Confusing a position vector with a displacement between two general points.

A position vector is always measured from the origin O.

Key takeaways

A position vector OA = a runs from the origin to the point.
AB = b − a (end minus start).
Distance from A to B is |b − a|.

Test yourself

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