Ext Maths · 7 Transformations and vectors

7.2 Vectors in two dimensions

Before you start

Reading and plotting coordinates in all four quadrants.
Adding and subtracting negative numbers, e.g. $3 - (- 1) = 4$ .
Solving a simple linear equation such as $p + 6 = 4$ .
From 7.1: a translation slides a shape without turning, flipping or resizing it.

Syllabus objectives covered (3)

Describe a translation using a vector represented by \binom{x}{y}, \overrightarrow{AB} or a.
Add and subtract vectors.
Multiply a vector by a scalar.

Part 1 of 5

What a column vector tells you

A vector is a movement: a distance in a particular direction. A column vector records that movement as two numbers stacked on top of each other.
$(\binom{x}{y})$
The top number is the movement across: positive means right, negative means left. The bottom number is the movement up or down: positive means up, negative means down. These are the same directions as the $x$ - and $y$ -axes.
So $(\binom{4}{- 3})$ means 4 right and 3 down, and $(\binom{- 2}{0})$ means 2 left with no vertical movement at all.
Write a vector as a column, never as coordinates like $(4, - 3)$ . Coordinates give a position; a vector gives a movement. An answer in coordinate form is a different kind of object, and you should not expect the mark for it.
You will see vectors written three ways, and they all mean the same kind of thing: a column vector $(\binom{x}{y})$ ; $\vec{A B}$ , the movement from point $A$ to point $B$ (the arrow shows the direction); or a single bold letter such as $𝐚$ . You cannot write bold by hand, so underline the letter instead: $a$ .
To find $\vec{A B}$ from coordinates, subtract the start from the end, top and bottom separately. For $A (1, 2)$ and $B (5, - 1)$ : across $5 - 1 = 4$ , up $- 1 - 2 = - 3$ .
$\vec{A B} = (\binom{4}{- 3})$
Going the other way reverses the direction but keeps the length, so every sign flips: $\vec{B A} = (\binom{- 4}{3}) = - \vec{A B}$ .

Part 2 of 5

Describing a translation

A translation moves every point of a shape by the same vector. That is why one column vector is enough to describe the whole transformation.
To find the vector, choose one vertex of the object and find the matching vertex on the image. Count across first, then up or down, going from the object to the image.
Match a vertex to its own image, count across then down: triangle $T$ is translated by $(\binom{4}{- 3})$ .
Take triangle $T$ with vertices $(1, 2)$ , $(3, 2)$ and $(1, 4)$ , moved to $T^{'}$ with vertices $(5, - 1)$ , $(7, - 1)$ and $(5, 1)$ . The vertex $(1, 2)$ goes to $(5, - 1)$ : that is 4 right and 3 down, so the vector is $(\binom{4}{- 3})$ .
The trap is counting the gap between the nearest edges of the two shapes. That distance means nothing. Always count from a vertex to its own image.
Check with a second vertex. $(3, 2)$ goes to $(7, - 1)$ : again 4 right and 3 down. If the two vertices disagree, you have matched the wrong vertices.
Direction matters. The question says which shape maps onto which. Counting from image back to object flips both signs and gives the wrong vector.
A full description needs two things: the word translation and the column vector. Write 'Translation by $(\binom{4}{- 3})$ '. Writing 'it moved 4 right and 3 down' does not use the language the mark is given for.
Going forwards, to translate a point by a vector, add the vector to its coordinates: $(1, 4)$ translated by $(\binom{4}{- 3})$ lands on $(1 + 4, 4 - 3) = (5, 1)$ .

Part 3 of 5

Adding and subtracting vectors

To add two column vectors, add the top numbers together and the bottom numbers together.
$(\binom{a}{b}) + (\binom{c}{d}) = (\binom{a + c}{b + d})$
The meaning behind this: $𝐚 + 𝐛$ is the single movement that does the same job as doing $𝐚$ and then doing $𝐛$ . Draw $𝐛$ starting where $𝐚$ finishes; the sum goes straight from the start of $𝐚$ to the end of $𝐛$ .
$𝐚$ then $𝐛$ , nose to tail, gets you to the same place as $𝐚 + 𝐛$ : $(\binom{4}{1}) + (\binom{1}{3}) = (\binom{5}{4})$ .
So $(\binom{4}{1}) + (\binom{1}{3}) = (\binom{5}{4})$ : 4 right then 1 more right is 5 right; 1 up then 3 more up is 4 up. The order does not matter, because $𝐚 + 𝐛 = 𝐛 + 𝐚$ .
To subtract, subtract top from top and bottom from bottom. Geometrically, $𝐚 - 𝐛$ means do $𝐚$ , then do $𝐛$ backwards, because $𝐚 - 𝐛 = 𝐚 + (- 𝐛)$ .
Negative numbers are where the marks go. Write the subtraction out with brackets before you simplify it:
$(\binom{3}{- 2}) - (\binom{- 1}{4}) = (\binom{3 - (- 1)}{- 2 - 4}) = (\binom{4}{- 6})$
The top is $3 - (- 1) = 4$ , not $2$ . Subtracting a negative adds.

Part 4 of 5

Multiplying a vector by a scalar

A scalar is an ordinary number, with size but no direction. To multiply a vector by a scalar, multiply both numbers in the column.
$k (\binom{x}{y}) = (\binom{k x}{k y})$
$3 (\binom{2}{- 1}) = (\binom{6}{- 3})$ . Same direction, three times as long: doing $(\binom{2}{- 1})$ three times in a row.
A fraction shrinks it: $\frac{1}{2} (\binom{6}{- 4}) = (\binom{3}{- 2})$ , same direction, half as long.
A negative scalar reverses the direction. $- 𝐚$ is $𝐚$ backwards, and $- 2 𝐚$ is twice as long as $𝐚$ and pointing the opposite way.
Whatever the scalar, $k 𝐚$ lies along the same line as $𝐚$ : it is parallel to $𝐚$ . You will use this in 7.4 to prove that lines are parallel.
For a combination such as $2 𝐚 - 3 𝐛$ , do the scalar multiplications first, write each result down, then add or subtract. Each vector you write down is working an examiner can give method credit for.

Part 5 of 5

Equal vectors and unknown numbers

Two column vectors are equal only when their top numbers are equal and their bottom numbers are equal.
So one vector equation gives you two ordinary equations: one from the top row and one from the bottom row. Solve them separately.
For example, $(\binom{2 x}{y + 1}) = (\binom{8}{- 3})$ gives $2 x = 8$ and $y + 1 = - 3$ , so $x = 4$ and $y = - 4$ .
When the equation contains scalars and sums, first simplify the left-hand side into a single column, then match rows. Finish by substituting your values back in to check.

Worked examples

Example 1. Triangle $T$ has vertices $(1, 1)$ , $(3, 1)$ and $(1, 4)$ . Triangle $U$ has vertices $(- 3, 3)$ , $(- 1, 3)$ and $(- 3, 6)$ . Describe fully the single transformation that maps triangle $T$ onto triangle $U$ .

Show the working

Both triangles are the same size and the same way round (the right angle is at the bottom-left vertex of each), so this is a translation.
Match a vertex of $T$ with its image in $U$ : $(1, 1)$ goes to $(- 3, 3)$ .
Across: $- 3 - 1 = - 4$ , so 4 left. Up: $3 - 1 = 2$ , so 2 up.
$(\binom{- 4}{2})$
Check with a second vertex: $(3, 1)$ goes to $(- 1, 3)$ , which is $- 1 - 3 = - 4$ across and $3 - 1 = 2$ up. The same vector, so the vertices were matched correctly.
State both parts of the description: the name of the transformation and the vector.

Answer: Translation by the vector $(\binom{- 4}{2})$ .

Example 2. $𝐚 = (\binom{3}{- 2})$ and $𝐛 = (\binom{- 1}{4})$ . Work out $2 𝐚 - 3 𝐛$ .

Show the working

Scalar multiplications first. Multiply both numbers in each column.
$2 𝐚 = (\binom{6}{- 4}), 3 𝐛 = (\binom{- 3}{12})$
Now subtract, writing the brackets in so that the negatives are handled deliberately.
$(\binom{6}{- 4}) - (\binom{- 3}{12}) = (\binom{6 - (- 3)}{- 4 - 12})$
Top: $6 + 3 = 9$ . Bottom: $- 4 - 12 = - 16$ .

Answer: $2 𝐚 - 3 𝐛 = (\binom{9}{- 16})$

Example 3. Find the value of $p$ and the value of $q$ when $(\binom{p}{- 1}) + 3 (\binom{2}{q}) = (\binom{4}{11})$ .

Show the working

Multiply the second vector by 3 first.
$3 (\binom{2}{q}) = (\binom{6}{3 q})$
Add, so that the left-hand side becomes a single column.
$(\binom{p + 6}{- 1 + 3 q}) = (\binom{4}{11})$
Top rows are equal: $p + 6 = 4$ , so $p = - 2$ .
Bottom rows are equal: $- 1 + 3 q = 11$ , so $3 q = 12$ and $q = 4$ .
Check: $(\binom{- 2}{- 1}) + (\binom{6}{12}) = (\binom{4}{11})$ . Correct.

Answer: $p = - 2$ and $q = 4$ .

Where people go wrong

Mistake: Writing a vector as coordinates, e.g. $(- 4, 2)$ .

Instead: Write it as a column: $(\binom{- 4}{2})$ .

Coordinates describe where a point is; a column vector describes a movement. They are different objects, even with the same numbers.

Mistake: Counting the translation from the image back to the object.

Instead: Count from the object to the image — from the shape the question maps from.

Going the wrong way flips both signs, giving $- 𝐯$ instead of $𝐯$ .

Mistake: Measuring the gap between the two shapes instead of matching vertices.

Instead: Pick one vertex and count to its own image, then check with a second vertex.

Every point moves by the same vector, but only corresponding points show it. The gap between edges is not a movement of anything.

Mistake: Getting $(\binom{3}{- 2}) - (\binom{- 1}{4}) = (\binom{2}{- 6})$ .

Instead: $3 - (- 1) = 4$ , so the answer is $(\binom{4}{- 6})$ .

Subtracting a negative number adds it. Writing the bracket $3 - (- 1)$ before simplifying prevents the slip.

Mistake: Multiplying only the top number by the scalar: $3 (\binom{2}{- 1}) = (\binom{6}{- 1})$ .

Instead: Multiply both: $(\binom{6}{- 3})$ .

A scalar stretches the whole movement, across and up equally. Changing one number would change the direction.

Check yourself

Describe in words the movement given by $(\binom{- 5}{2})$ .
Answer
5 units left and 2 units up.
$A$ is $(2, - 1)$ and $B$ is $(- 3, 4)$ . Write $\vec{A B}$ and $\vec{B A}$ as column vectors.
Answer
$\vec{A B} = (\binom{- 5}{5})$ (end minus start), and $\vec{B A} = (\binom{5}{- 5})$ .
The point $(5, 2)$ is translated by $(\binom{- 3}{- 4})$ . Where does it end up?
Answer
$(5 - 3, 2 - 4) = (2, - 2)$ .
What two things must a full description of a translation contain?
Answer
The word 'translation' and the column vector of the translation.
$𝐚 = (\binom{4}{1})$ and $𝐛 = (\binom{- 2}{3})$ . Work out $𝐚 + 𝐛$ and $𝐚 - 𝐛$ .
Answer
$𝐚 + 𝐛 = (\binom{2}{4})$ and $𝐚 - 𝐛 = (\binom{6}{- 2})$ .
$𝐜 = (\binom{- 6}{9})$ . Work out $- 2 𝐜$ and $\frac{1}{3} 𝐜$ . Which one points the same way as $𝐜$ ?
Answer
$- 2 𝐜 = (\binom{12}{- 18})$ and $\frac{1}{3} 𝐜 = (\binom{- 2}{3})$ . Only $\frac{1}{3} 𝐜$ points the same way; the negative scalar reverses $- 2 𝐜$ .

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. Triangle $P$ has vertices $(1, 2)$ , $(4, 2)$ and $(4, 4)$ . Triangle $Q$ has vertices $(3, - 3)$ , $(6, - 3)$ and $(6, - 1)$ .

(a) Describe fully the single transformation that maps triangle $P$ onto triangle $Q$ . 2 marks No calculator

Mark scheme

B1 translation
B1 $(\binom{2}{- 5})$

(b) Triangle $P$ is translated by the vector $(\binom{- 4}{1})$ to give triangle $R$ . Write down the coordinates of the three vertices of triangle $R$ . 2 marks No calculator

Mark scheme

B2 $(- 3, 3)$ , $(0, 3)$ , $(0, 5)$ ; B1 for two correct vertices

(c) Write down the column vector of the translation that maps triangle $R$ onto triangle $P$ . 1 mark No calculator

Mark scheme

B1 $(\binom{4}{- 1})$

Question 2. $𝐚 = (\binom{5}{- 2})$ and $𝐛 = (\binom{- 3}{4})$ .

(a) Work out $𝐚 + 𝐛$ . 1 mark No calculator

Mark scheme

B1 $(\binom{2}{2})$

(b) Work out $3 𝐚 - 2 𝐛$ . 2 marks No calculator

Mark scheme

M1 $(\binom{15}{- 6})$ or $(\binom{- 6}{8})$ seen
A1 $(\binom{21}{- 14})$

(c) $m 𝐚 + n 𝐛 = (\binom{- 1}{6})$ . Find the value of $m$ and the value of $n$ . 3 marks No calculator

Mark scheme

M1 both equations formed: $5 m - 3 n = - 1$ and $- 2 m + 4 n = 6$ oe
M1 correct method to eliminate one unknown
A1 $m = 1$ and $n = 2$

(d) $𝐝 = (\binom{- 10}{4})$ . Explain how the vector $𝐝$ is related to the vector $𝐚$ . 2 marks No calculator

Mark scheme

B1 $𝐝 = - 2 𝐚$ oe
B1 twice as long as $𝐚$ and in the opposite direction oe

How this is examined

Describing a translation with both the word and the vector: A 2-mark 'Describe fully the single transformation' after two shapes on a grid or given by their vertices One mark for the word 'translation', one for the correct column vector counted from object to image, e.g. $(\binom{- 4}{2})$ . A vector written as coordinates, or 'moved left and up', is not what the marks are given for.
Working out a combination like $3 𝐚 - 2 𝐛$ : A 1- or 2-mark 'Work out' with two column vectors given in the stem A correct scalar multiple such as $3 𝐚$ written down as a column earns method credit; the final single column vector, with signs right, earns the accuracy mark.
Finding unknowns by matching top rows and bottom rows: A 2- or 3-mark 'Find the value of' with a vector equation such as $m 𝐚 + n 𝐛 = (\binom{x}{y})$ Writing the two separate equations (one from each row) earns method credit, solving them correctly earns the rest. Substituting back is a free check.
Reversing a translation: A 1-mark 'Write down' the vector that maps the image back onto the object Both signs flipped: the reverse of $(\binom{- 4}{1})$ is $(\binom{4}{- 1})$ .

Words to know

Vector: A quantity with both size and direction, such as a movement. Written as $(\binom{x}{y})$ , $\vec{A B}$ or a bold letter $𝐚$ (underlined by hand).
Column vector: A vector written as $(\binom{x}{y})$ : $x$ across (right positive), $y$ up (up positive).
Scalar: An ordinary number with size but no direction. Multiplying a vector by a scalar changes its length, and reverses it if the scalar is negative.
Translation: A transformation that moves every point of a shape by the same vector, with no turning, reflecting or resizing.
Object and image: The shape before a transformation (object) and after it (image).

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