Position Vectors

Use the vector from the origin to name a point, find the vector between two points, find a new point, and find a midpoint.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1, Edexcel IGCSE Maths 4MA1 · 46 real past-paper questions below

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What a Position Vector Is

A position vector starts at the origin O and ends at a point. Its numbers are the same as the point's coordinates.

Steps

1
A(3, −2): $O A = a = (3 - 2)$
2
B(0, 5): $O B = b = (05)$

Key idea

The small letter goes with the capital: point A has position vector $a$ , point B has $b$ .

The Vector Between Two Points

The most common question: find $A B$ from the two points. Go back from A to O, then out from O to B.

Worked example

1
$a = (- 2 3), b = (41)$ — A is (−2, 3) and B is (4, 1). A position vector has the same numbers as the point, so write each as a column.
2
$A B = A O + O B = - a + b$ — Walk from A back to O, then out to B. Going back along a is the opposite of a, so it gets a minus.
3
$A B = b - a$ — Write the positive one first. It is end point minus start point, and you can use it every time.
4
$A B = (41) - (- 2 3)$ — Put in the two columns. Keep the brackets round the negative number so its sign is not lost.
5
$4 - (- 2) = 6$ — Do the top numbers. Taking away a negative adds, so this is 4 + 2.
6
$1 - 3 = - 2$ — Do the bottom numbers.
7
$A B = (6 - 2)$ — Write the two answers back into one column.

Rule

$A B = b - a$ : end minus start.

Common mistake

Working out a − b. That is the vector from B to A, with both signs wrong.

Finding a Point From a Vector

The 3-mark question gives a point and some vectors, and asks for the coordinates of a new point. Start at the point you know and add each move.

Worked example

1
$A (2, 5), A B = (4 - 3)$ — You know where A is and the move from A to B. You want the coordinates of B.
2
$O B = O A + A B$ — To reach B, walk from O to A, then take the move from A to B.
3
$O B = (25) + (4 - 3)$ — OA has the same numbers as the point A. Put the two columns in.
4
$2 + 4 = 6$ — Add the top numbers.
5
$5 + (- 3) = 2$ — Add the bottom numbers. Adding −3 is the same as taking 3 away.
6
$O B = (62) \Rightarrow B (6, 2)$ — A position vector has the same numbers as its point, so this is the point B.
7
$B C = (- 1 6)$ — One more move. It starts at B, where the last one finished.
8
$O C = O B + B C = (62) + (- 1 6)$ — Walk from O to B, then take the move from B to C.
9
$6 + (- 1) = 5$ — Add the top numbers.
10
$2 + 6 = 8$ — Add the bottom numbers.
11
$O C = (58) \Rightarrow C (5, 8)$ — Write the point with the same numbers.

Common mistake

Subtracting the move. You add the vector to go from A to B. Subtracting takes you the other way, to (−2, 8).

Working Backwards to the Start

Sometimes you know where the vector ends and need the point it started from. Walk the other way: take the move away.

Q is the point with $O Q = (14)$ and $P Q = (- 3 5)$ . Find the coordinates of P.

Worked example

1
$O Q = O P + P Q$ — Going from O to P and then from P to Q reaches Q. You know the end and the move, but not the start.
2
$O P = O Q - P Q$ — Take PQ away from both sides to leave OP on its own.
3
$O P = (14) - (- 3 5)$ — Put in the columns, with brackets round the negative number.
4
$1 - (- 3) = 4$ — Subtract the top numbers. Taking away −3 adds 3.
5
$4 - 5 = - 1$ — Subtract the bottom numbers.
6
$O P = (4 - 1) \Rightarrow P (4, - 1)$ — The position vector has the same numbers as the point.
7
$(4 - 1) + (- 3 5) = (14)$ — Check it: starting at P and adding PQ must land on Q, and it does.

Rule

You know the start: add the move. You know the end: subtract the move. In $B A$ the end is A and the start is B, so read the letters in order.

Key idea

Show the line $O P = O Q - P Q$ . Most of the marks are for the working, so a correct line with a slip in the arithmetic still scores.

The Midpoint

The midpoint M of AB is exactly halfway. Its position vector is half of $a + b$ : add the two vectors and halve.

Formula

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

Worked example

1
$a = (- 2 3), b = (41)$ — Same points, A(−2, 3) and B(4, 1). Write their position vectors as columns.
2
$a + b = (- 2 + 4 3 + 1)$ — Add the top numbers together and the bottom numbers together. Halving this sum finds the number halfway between.
3
$a + b = (24)$ — Work out each sum.
4
$m = \frac{1}{2} (24)$ — Halve it. The midpoint is exactly halfway between A and B.
5
$m = (12) \Rightarrow M (1, 2)$ — Half of 2 is 1 and half of 4 is 2. The position vector gives the coordinates of M.

Common mistake

Halving b − a. That gives half the move from A to B, not the position of M.

Which Method? Read the Question

Look for the origin O, and for which point is the start.

You see	You do
Position vector of a point	same numbers as its coordinates
Find AB from two points	b − a, end minus start
Coordinates of a new point	start point plus each move
Coordinates of the start point	end point minus the move
Midpoint	half of a + b
How far apart?	Magnitude of a Vector
Points in a shape with a and b	Expressing Vectors in Terms of a and b

Key idea

Draw the arrows on a quick sketch. It shows you which point is the start and which is the end.

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