Hard GCSE Vector notation and translations Questions

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 $\begin{pmatrix} -6 \\ -9 \end{pmatrix}$.

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 $A(-2, -3)$ from $A'(-5, 4)$ gives the column vector $\begin{pmatrix} -3 \\ 7 \end{pmatrix}$.

Check the same vector works for the other two vertices

The same vector $\begin{pmatrix} -3 \\ 7 \end{pmatrix}$ moves $B$ and $C$ to their images too, so it describes the whole transformation.

Rule out alternative number 1

A translation by $\begin{pmatrix} 7 \\ -3 \end{pmatrix}$ would send $A(-2, -3)$ to $(5, -6)$, but the image of $A$ is $A'(-5, 4)$ — so that vector is wrong.

Rule out alternative number 2

A translation by $\begin{pmatrix} 3 \\ -7 \end{pmatrix}$ would send $A(-2, -3)$ to $(1, -10)$, but the image of $A$ is $A'(-5, 4)$ — so that vector is wrong.

Rule out alternative number 3

A translation by $\begin{pmatrix} -3 \\ -7 \end{pmatrix}$ would send $A(-2, -3)$ to $(-5, -10)$, but the image of $A$ is $A'(-5, 4)$ — so that vector is wrong.

Rule out alternative number 4

A translation by $\begin{pmatrix} 3 \\ 7 \end{pmatrix}$ would send $A(-2, -3)$ to $(1, 4)$, but the image of $A$ is $A'(-5, 4)$ — 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 3 left and then 7 up, 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 $\begin{pmatrix} -3 \\ 7 \end{pmatrix}$.

Write the vector as a column vector

Subtracting $A(4, -5)$ from $B(-2, 1)$ gives $\vec{AB} = \begin{pmatrix} -6 \\ 6 \end{pmatrix}$.

Draw the right-angled triangle the vector makes with the grid

The vector is the hypotenuse of a right-angled triangle whose legs are $6$ across and $6$ up, so Pythagoras will give its length.

Write down Pythagoras for the magnitude

The magnitude of a vector is the length of the arrow, and that is the hypotenuse of the right-angled triangle formed by its two components.

Square the top number

Squaring $-6$ gives $36$. Squaring always gives a positive result, which is why the direction cannot change the length.

Square the bottom number

And $6$ squared is $36$.

Add the two squares

The sum of the squares is $72$, which is the square of the length.

Take the square root

The magnitude is $\sqrt{72}$.

Simplify the surd

$72$ has the square factor $36$, so the surd simplifies to $6\sqrt{2}$.

Note that the signs have disappeared

Squaring destroys the minus signs, so a vector and its reverse have exactly the same magnitude — direction is lost, only length remains.

Give a decimal value only as a check

To one decimal place the length is about $8.5$ units, but the exact answer $6\sqrt{2}$ is what should be written down unless the question asks for a rounded value.

Note that the reverse vector has the same magnitude

Reversing a vector changes both signs but not its length, so the magnitude of the reverse vector is the same.

Note the mistake to avoid

The magnitude is not the sum of the two numbers, and it is not the bigger of them. Both components must be squared, added, then square rooted.

Sense check the size of the answer

The length must be more than the longer leg and less than the two legs added together, and $8.5$ sits between them.

Summarise the method

Square both numbers, add them, take the square root, then simplify the surd. Leave the answer exact.

State the exact magnitude

The magnitude is $6\sqrt{2}$ units, in exact form.

Write the vector as a column vector

The vector is given as $\begin{pmatrix} -9 \\ 12 \end{pmatrix}$.

Draw the right-angled triangle the vector makes with the grid

The vector is the hypotenuse of a right-angled triangle whose legs are $9$ across and $12$ up, so Pythagoras will give its length.

Write down Pythagoras for the magnitude

The magnitude of a vector is the length of the arrow, and that is the hypotenuse of the right-angled triangle formed by its two components.

Square the top number

Squaring $-9$ gives $81$. Squaring always gives a positive result, which is why the direction cannot change the length.

Square the bottom number

And $12$ squared is $144$.

Add the two squares

The sum of the squares is $225$, which is the square of the length.

Take the square root

The magnitude is $\sqrt{225}$.

Simplify the surd

$225$ is a square number, so the root is the whole number $15$.

Note that the signs have disappeared

Squaring destroys the minus signs, so a vector and its reverse have exactly the same magnitude — direction is lost, only length remains.

Give a decimal value only as a check

To one decimal place the length is about $15.0$ units, but the exact answer $15$ is what should be written down unless the question asks for a rounded value.

Note that the reverse vector has the same magnitude

Reversing a vector changes both signs but not its length, so the magnitude of the reverse vector is the same.

Note the mistake to avoid

The magnitude is not the sum of the two numbers, and it is not the bigger of them. Both components must be squared, added, then square rooted.

Sense check the size of the answer

The length must be more than the longer leg and less than the two legs added together, and $15.0$ sits between them.

Summarise the method

Square both numbers, add them, take the square root, then simplify the surd. Leave the answer exact.

State the exact magnitude

The magnitude is $15$ units, in exact form.

Write the vector as a column vector

Subtracting $A(-3, -2)$ from $B(3, 4)$ gives $\vec{AB} = \begin{pmatrix} 6 \\ 6 \end{pmatrix}$.

Draw the right-angled triangle the vector makes with the grid

The vector is the hypotenuse of a right-angled triangle whose legs are $6$ across and $6$ up, so Pythagoras will give its length.

Write down Pythagoras for the magnitude

The magnitude of a vector is the length of the arrow, and that is the hypotenuse of the right-angled triangle formed by its two components.

Square the top number

Squaring $6$ gives $36$. Squaring always gives a positive result, which is why the direction cannot change the length.

Square the bottom number

And $6$ squared is $36$.

Add the two squares

The sum of the squares is $72$, which is the square of the length.

Take the square root

The magnitude is $\sqrt{72}$.

Simplify the surd

$72$ has the square factor $36$, so the surd simplifies to $6\sqrt{2}$.

Note that the signs have disappeared

Squaring destroys the minus signs, so a vector and its reverse have exactly the same magnitude — direction is lost, only length remains.

Give a decimal value only as a check

To one decimal place the length is about $8.5$ units, but the exact answer $6\sqrt{2}$ is what should be written down unless the question asks for a rounded value.

Note that the reverse vector has the same magnitude

Reversing a vector changes both signs but not its length, so the magnitude of the reverse vector is the same.

Note the mistake to avoid

The magnitude is not the sum of the two numbers, and it is not the bigger of them. Both components must be squared, added, then square rooted.

Sense check the size of the answer

The length must be more than the longer leg and less than the two legs added together, and $8.5$ sits between them.

Summarise the method

Square both numbers, add them, take the square root, then simplify the surd. Leave the answer exact.

State the exact magnitude

The magnitude is $6\sqrt{2}$ units, in exact form.