A vector in component form shows how far it moves along x and along y. The vector from A to B is always AB = OB − OA: the end point minus the start point.
This lesson is part of vectors. The next lesson, finding magnitude and direction, uses the vectors you build here.
How do I write a vector?
The letters i and j are unit vectors along the x-axis and y-axis. A vector with 3 units along x and 4 units down along y can be written in two ways.
| Form | Example |
|---|---|
| Column | (3, −4) |
| Unit-vector | 3i − 4j |
You can add, subtract and scale vectors component by component. For a = 2i + 3j and b = i − 4j:
2a − 3b = (4i + 6j) − (3i − 12j) = i + 18j.
Worked example 1: the vector between two points
A is the point (2, 5) and B is (8, −3). Find AB.
The position vectors are OA = 2i + 5j and OB = 8i − 3j. Then AB = OB − OA = (8 − 2)i + (−3 − 5)j = 6i − 8j.
Check the direction with a picture: going from A to B, x increases by 6 and y falls by 8. The answer has a positive i part and a negative j part, which matches.
Worked example 2: the midpoint
P is (−1, 4) and Q is (5, −2). Find PQ and the midpoint M of PQ.
PQ = OQ − OP = (5 − (−1))i + (−2 − 4)j = 6i − 6j. The midpoint has position vector OM = ½(OP + OQ) = ½[(−i + 4j) + (5i − 2j)] = ½(4i + 2j) = 2i + j, so M is the point (2, 1).
The mistake that costs marks
The common slip is to subtract in the wrong order and write AB = OA − OB. The result is the vector BA, which points the opposite way.
| Step | Wrong | Right |
|---|---|---|
| Formula | AB = OA − OB | AB = OB − OA |
| Calculation (A(2, 5), B(8, −3)) | (2 − 8)i + (5 − (−3))j | (8 − 2)i + (−3 − 5)j |
| Result | −6i + 8j, which is BA | 6i − 8j |
Say the rule aloud: “end minus start”. If the question asks for AB, B is the end.
Check yourself
P is the point (−1, 4) and Q is the point (5, −2). Write QP in i, j form and say how it compares with PQ.
Answer
QP = OP − OQ = (−1 − 5)i + (4 − (−2))j = −6i + 6j.
It is the negative of PQ = 6i − 6j. Reversing the journey gives the same length in the opposite direction, so QP = −PQ.
What to study next
Next, measure a vector and find its direction in finding magnitude and direction. Then test the whole chapter with the vectors practice set. The units and significant-figure checker helps with the final rounding.
If you want a teacher to work through vector questions with you, see online one-to-one Additional Mathematics tuition.