Use position vectors and find unit vectors
Represent a point from the origin, find a displacement by subtracting position vectors, and scale a nonzero vector to unit length.
Section 13 · Vectors in two dimensions
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
If OA = a and OB = b, what is AB?
What you will learn
- Use a position vector to describe a point from the origin
- Find a vector between points from their position vectors
- Find the unit vector in the direction of a nonzero vector
Position vectors start at the origin
The position vector OA gives the coordinates of point A when O is the origin. If OA = a and OB = b, then AB = b − a: move from O to B and subtract the displacement from O to A.
Keep the direction clear: AB = −BA. A point's position vector and the vector between two points are different unless the start point is O.
A unit vector has length one
For a nonzero vector v, its magnitude is |v| = √(v₁² + v₂²). The unit vector in its direction is v/|v|; this divides each component by the same positive magnitude.
The direction is unchanged because the scale factor is positive. The zero vector has no direction, so it has no unit vector.
Pause and explain
How do you form the unit vector in the direction of (3, 4)?
Worked example
A has position vector (2, −1) and B has position vector (7, 3). Find AB and its unit vector.
Show the worked solution
- Subtract the position vectors: AB = (7 − 2, 3 − (−1)) = (5, 4).
- Its magnitude is √(5² + 4²) = √41.
- Divide each component by √41 to obtain a vector of length one in the same direction.
Answer AB = (5, 4); the unit vector is (5/√41, 4/√41).
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A = (1, −2), B = (5, 1). Find AB and the unit vector in its direction.
Give me a hint
First subtract positions; then divide by the displacement magnitude.
Compare my reasoning
- AB = (5 − 1, 1 − (−2)) = (4, 3).
- Its magnitude is √(4² + 3²) = 5.
- Divide by 5.
AB = (4, 3); unit vector (4/5, 3/5).
Look for these in your work
- Displacement direction correct
- Magnitude calculated
- Unit vector has length one
Find the unit vector in the direction of v = (−5, 12).
Give me a hint
Find the magnitude, then divide both components by it.
Compare my reasoning
- |v| = √(25 + 144) = 13.
- Divide each component by 13.
- The negative horizontal sign remains because the vector points left.
(−5/13, 12/13).
Look for these in your work
- Magnitude 13 found
- Both components scaled equally
- Direction signs preserved
A rover moves from A(−3, 2) to B(3, 10). Find its displacement and unit direction vector.
Give me a hint
Use end position minus start position, then normalise the result.
Compare my reasoning
- AB = (3 − (−3), 10 − 2) = (6, 8).
- The magnitude is √(36 + 64) = 10.
- Divide each component by 10.
Displacement (6, 8); unit direction (3/5, 4/5).
Look for these in your work
- Coordinates subtracted in order
- Distance of displacement found
- Unit vector simplified
Common mistakes
- Using OA − OB to find AB
- Dividing a vector by its squared magnitude
- Changing the direction by dividing by a negative value
- Trying to find a unit vector for the zero vector
What can you explain now?
What is the magnitude of (3, 4)?
Compare with the explanation
5
√(3² + 4²) = 5.
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Make it stick
How are position vectors, displacements and unit vectors related?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.