GCSE Vector notation and translations Practice Questions

Write the translation as a coordinate rule

Translating by $\begin{pmatrix} 4 \\ 3 \end{pmatrix}$ adds $4$ to every $x$-coordinate and $3$ to every $y$-coordinate.

Apply the rule to P

Adding the vector to the coordinates of $P(2, 1)$ gives $P'(6, 4)$.

State the coordinates of the image

The image of $P(2, 1)$ under the translation $\begin{pmatrix} 4 \\ 3 \end{pmatrix}$ is $P'(6, 4)$.

Use the rule that the vector is the image minus the object

The vector of a translation is found by subtracting the starting point from the finishing point.

Subtract the coordinates to get the column vector

The translation is $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$: 4 right and then 2 up.

State the column vector

The column vector that maps $P$ onto $Q$ is $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$.

Write down the two points

The translation has to take $P(3, -1)$ exactly onto $Q(-2, 3)$.

Use the rule that the vector is the image minus the object

The vector of a translation is found by subtracting the starting point from the finishing point.

Work out the top number

The $x$-coordinate changes by $-5$, so the top number is $-5$.

Work out the bottom number

The $y$-coordinate changes by $4$, so the bottom number is $4$.

Rule out alternative number 1

Translating $P(3, -1)$ by $\begin{pmatrix} 4 \\ -5 \end{pmatrix}$ lands on $(7, -6)$, not on $Q(-2, 3)$ — so that vector is not the answer.

State the column vector

The column vector that maps $P$ onto $Q$ is $\begin{pmatrix} -5 \\ 4 \end{pmatrix}$.

Write down the two points

The translation has to take $P(5, -2)$ exactly onto $Q(-1, 6)$.

Use the rule that the vector is the image minus the object

The vector of a translation is found by subtracting the starting point from the finishing point.

Work out the top number

The $x$-coordinate changes by $-6$, so the top number is $-6$.

Work out the bottom number

The $y$-coordinate changes by $8$, so the bottom number is $8$.

Subtract the coordinates to get the column vector

The translation is $\begin{pmatrix} -6 \\ 8 \end{pmatrix}$: 6 left and then 8 up.

Rule out alternative number 1

Translating $P(5, -2)$ by $\begin{pmatrix} 8 \\ -6 \end{pmatrix}$ lands on $(13, -8)$, not on $Q(-1, 6)$ — so that vector is not the answer.

Rule out alternative number 2

Translating $P(5, -2)$ by $\begin{pmatrix} 6 \\ -8 \end{pmatrix}$ lands on $(11, -10)$, not on $Q(-1, 6)$ — so that vector is not the answer.

Rule out alternative number 3

Translating $P(5, -2)$ by $\begin{pmatrix} -6 \\ -8 \end{pmatrix}$ lands on $(-1, -10)$, not on $Q(-1, 6)$ — so that vector is not the answer.

Rule out alternative number 4

Translating $P(5, -2)$ by $\begin{pmatrix} 6 \\ 8 \end{pmatrix}$ lands on $(11, 6)$, not on $Q(-1, 6)$ — so that vector is not the answer.

State the column vector

The column vector that maps $P$ onto $Q$ is $\begin{pmatrix} -6 \\ 8 \end{pmatrix}$.

Write the object and the image side by side

Matching vertices are $A$ with $A'$, $B$ with $B'$ and $C$ with $C'$. The description has to be read from those pairs.

Check the shape is the same size and the same way up

The image is congruent to the object and has not been turned or flipped, so the transformation is a translation — the only thing left to find is the vector.

Subtract to find the translation vector

Subtracting $A(3, 4)$ from $A'(-3, -5)$ gives the column vector $\begin{pmatrix} -6 \\ -9 \end{pmatrix}$.

Check the same vector works for the other two vertices

The same vector $\begin{pmatrix} -6 \\ -9 \end{pmatrix}$ moves $B$ and $C$ to their images too, so it describes the whole transformation.

Rule out alternative number 1

A translation by $\begin{pmatrix} -9 \\ -6 \end{pmatrix}$ would send $A(3, 4)$ to $(-6, -2)$, but the image of $A$ is $A'(-3, -5)$ — so that vector is wrong.

Rule out alternative number 2

A translation by $\begin{pmatrix} 6 \\ 9 \end{pmatrix}$ would send $A(3, 4)$ to $(9, 13)$, but the image of $A$ is $A'(-3, -5)$ — so that vector is wrong.

Rule out alternative number 3

A translation by $\begin{pmatrix} -6 \\ 9 \end{pmatrix}$ would send $A(3, 4)$ to $(-3, 13)$, but the image of $A$ is $A'(-3, -5)$ — so that vector is wrong.

Rule out alternative number 4

A translation by $\begin{pmatrix} 6 \\ -9 \end{pmatrix}$ would send $A(3, 4)$ to $(9, -5)$, but the image of $A$ is $A'(-3, -5)$ — so that vector is wrong.

Explain why only one vector can be right

A translation is fixed completely by what it does to a single point, so exactly one of the five vectors can map the object onto the image.

Say how the image is related to the object

A translation slides every point by the same vector, so the image is the same shape, the same size and the same way up as the object: the two triangles are congruent.

Identify any invariant points

A translation by a vector that is not zero moves every single point, so no point of the shape stays where it was.

Note the mistake to avoid

The top number moves the shape left or right and the bottom number moves it up or down. Swapping them, or losing a minus sign, is the commonest slip in the whole topic.

Sense check the vector against the picture

On the grid the triangle really has moved 6 left and then 9 down, which matches the signs of the vector.

Summarise how to describe a translation fully

A full description of a translation names the transformation and gives the column vector. Nothing else is needed — and nothing else will earn the marks.

State the full description

The single transformation is a translation by the column vector $\begin{pmatrix} -6 \\ -9 \end{pmatrix}$.