Column Vectors and Translations

Write a move as a column vector, read one from a grid, reverse it, and join two moves to find a third.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 9 real past-paper questions below

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What a Column Vector Is

A vector is a move. It has a size and a direction. A column vector writes the move as two numbers, one on top of the other.

Formula

(3 - 2)

Number	Meaning
Top	across: plus is right, minus is left
Bottom	up or down: plus is up, minus is down

You will see three names for a vector: $A B$ (the move from A to B), a bold letter like $a$ , or the column itself. By hand, underline the letter.

Key idea

A vector has no fixed place. The same move drawn anywhere on the grid is the same vector.

Key idea

A vector would make a funny tour guide. It tells you exactly how far to walk and in which direction, but it has no idea where you are starting from.

Reading a Vector From a Grid

The 1-mark question: write the vector from one point to another. Start at the first point. Count across, then up or down.

Worked example

1
$A (1, 4) to B (6, 1)$ — $A B$ means start at A and finish at B. The top number is the move across, the bottom number is the move up or down.
2
$6 - 1 = 5$ — Across is the change in x: end x minus start x. The answer is plus, so the move is 5 to the right.
3
$1 - 4 = - 3$ — Up or down is the change in y: end y minus start y. The answer is minus, so the move is 3 down.
4
$A B = (5 - 3)$ — Write the across number on top and the up or down number underneath.

The same method works when the coordinates are negative. Watch the signs.

Worked example

1
$C (- 2, 3) to D (4, - 1)$ — A second one, this time with negative coordinates. Put every negative number in brackets.
2
$4 - (- 2) = 6$ — End x minus start x. Taking away a negative number adds, so 4 − (−2) = 4 + 2.
3
$- 1 - 3 = - 4$ — End y minus start y. Taking 3 away from −1 goes further below zero, so the move is 4 down.
4
$C D = (6 - 4)$ — Across on top, down underneath. Check: 6 right and 4 down is the way D sits from C.

Key idea

To draw a vector on a grid, start at any point. Go across first, then up or down, and put the arrowhead at the end.

Common mistake

Counting from B back to A. That gives every sign the wrong way round.

Going Backwards: the Negative Vector

The move from B to A is the same move done backwards. Flip the sign of both numbers.

Worked example

1
$A B = (5 - 3)$ — This is the vector from the last card: 5 across and 3 down.
2
$B A = - A B$ — BA goes from B back to A, which undoes AB. Undoing a move is the same as putting a minus in front.
3
$- (5) = - 5$ — Apply the minus to the top number. 5 to the right becomes 5 to the left.
4
$- (- 3) = 3$ — Apply the minus to the bottom number. A minus before a minus makes a plus, so 3 down becomes 3 up.
5
$B A = (- 5 3)$ — Put both new numbers back into one column. Both signs have flipped.

Rule

If $a = (5 - 3)$ , then $- a = (- 5 3)$ . Same length, opposite direction.

Common mistake

Flipping only one number. That gives a different direction, not the reverse.

Joining Two Moves: Find AC

The 2-mark question gives $A B$ and $C B$ , and asks for $A C$ . Go from A to B, then from B to C.

Worked example

1
$A B = (4 - 1), C B = (- 2 5)$ — We are given AB and CB, and we want AC.
2
$A C = A B + B C$ — Go from A to B, then from B to C. That journey ends at the same place as going from A straight to C.
3
$B C = - C B$ — We were given CB, which points the wrong way for this route. Reverse it.
4
$B C = (- (- 2) - (5)) = (2 - 5)$ — Reversing flips both signs: a minus before −2 makes +2, and a minus before 5 makes −5.
5
$4 + 2 = 6$ — Add the tops of AB and BC.
6
$- 1 + (- 5) = - 6$ — Add the bottoms. Adding −5 is the same as taking 5 away.
7
$A C = (6 - 6)$ — Put both answers in one column.

Common mistake

Adding CB as it is. CB points the wrong way for this route, so reverse it first. Adding it unchanged gives (2, 4) as a column, which is wrong.

Sometimes both vectors you are given start from the same letter. Then you do not need to reverse twice.

Worked example

1
$B A = (- 5 4), B C = (91)$ — Now both given moves start at B. We still want AC.
2
$A C = A B + B C$ — Go from A to B, then from B to C.
3
$A B = - B A = (5 - 4)$ — AB is the reverse of the given BA: flip both signs of (−5, 4).
4
$5 + 9 = 14$ — Add the tops of AB and BC.
5
$- 4 + 1 = - 3$ — Add the bottoms of AB and BC.
6
$A C = (14 - 3)$ — Put both answers in one column.

Key idea

A shortcut when both start at B: $A C = B C - B A$ . Here that is $(9 - (- 5), 1 - 4) = (14, - 3)$ , the same answer.

Moving a Point by a Vector

Many questions give you a point and a vector, and ask for another point. Use this whenever you see a vector joined to a point.

Formula

Q = (x_{P} + p, y_{P} + q) when P Q = (p q)

$P$ is the start point $(x_{P}, y_{P})$ , $Q$ is the end point, and $p$ and $q$ are the top and bottom of the vector. The top number moves x and the bottom number moves y. To find the start point instead, take the vector away: $P = (x_{Q} - p, y_{Q} - q)$ .

Moving a whole shape by a vector is the same job: add the vector to every corner.

Worked example

1
$P (- 4, 3), P Q = (7 - 5)$ — We know the start point and the move. We want the end point Q.
2
$- 4 + 7 = 3$ — Add the top number to the x-coordinate of P. Plus 7 means 7 to the right.
3
$3 + (- 5) = - 2$ — Add the bottom number to the y-coordinate of P. Adding −5 means 5 down.
4
$Q = (3, - 2)$ — Write the answer as a point, with a comma and round brackets, not as a column.

Now the other way round. The unknown point is the start of the vector, so you must go backwards.

Worked example

1
$F E = (2 - 4), E (6, 1)$ — FE goes from F to E. We know where it ends but not where it starts.
2
$F = E - F E$ — Undo the move: take the vector away from the end point.
3
$6 - 2 = 4$ — Take the top number away from the x-coordinate of E.
4
$1 - (- 4) = 1 + 4 = 5$ — Take the bottom number away from the y-coordinate. A minus before a minus makes a plus.
5
$F = (4, 5)$ — Check by moving forward: (4 + 2, 5 + (−4)) = (6, 1), which is E.

Common mistake

Adding the vector when the unknown point is the first letter. In FE the unknown is F, the start, so you subtract.

Key idea

Write the sum for each coordinate, such as −4 + 7 = 3. On a 2-mark question, one correct coordinate or one correct line of working usually earns a mark even if the other slips.

Which Method? Read the Question

Look at what you are given and which way the arrow goes.

You see	You do
Two points or a grid	count across, then up or down
BA, or −a	flip both signs
AB and CB, find AC	route A to B to C, reverse CB
Move a shape by a vector	Translation
A point and a vector, find the other point	add the vector to the start; take it away to find the start
Add, subtract, missing numbers	Adding and Subtracting Vectors
How long is the vector?	Magnitude of a Vector

Key idea

Always read the letters in order. The first letter is the start, the second is the end.

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