Transformations and vectors

Geometric tiles are built by reflecting, rotating and translating shapes.

A transformation 变换 changes the position or size of a shape. The original is the object and the result is the image. There are four types. When asked to describe one, you must name the type and give all the details below.

Reflection

A reflection 反射 flips the shape over a mirror line 对称轴. Each image point is the same distance from the line as the object point, on the other side.

• To describe it, give the equation of the mirror line (for Core, a horizontal or vertical line; Extended allows any line such as $y = x$).

Worked example. Reflect the point $(3, 2)$ in the $y$-axis. Only the sign of $x$ changes: the image is $(-3, 2)$. (In the $x$-axis it would be $(3, -2)$.)

A reflection flips the shape over a mirror line (here the $y$-axis); each point and its image are the same distance from the line.

Rotation

A rotation 旋转 turns the shape about a fixed point, the centre 中心 of rotation, through multiples of $90^{\circ}$.

• To describe it, give the centre, the angle, and the direction (clockwise or anticlockwise).

Worked example. Rotate $(3, 1)$ by $90^{\circ}$ anticlockwise about the origin. The rule is $(x, y) \to (-y, x)$, so the image is $(-1, 3)$.

A rotation turns the shape about a fixed centre — here $90^\circ$ anticlockwise about the origin.

Enlargement

An enlargement 放大 changes the size by a scale factor 比例因子 $k$, measured from a fixed centre. Each distance from the centre is multiplied by $k$.

• To describe it, give the centre and the scale factor. For Extended, $k$ may be fractional (the shape gets smaller) or negative (the image appears on the other side of the centre).

Worked example. Enlarge $(1, 2)$ from the origin by scale factor $2$. Multiply both coordinates: the image is $(2, 4)$.

An enlargement multiplies every distance from the centre by the scale factor (here $2$).

Translation

A translation 平移 slides the shape with no turning, by a vector 向量 written as a column vector 列向量 $\begin{pmatrix} x \\ y \end{pmatrix}$ ($x$ across, $y$ up).

Worked example. Translate $(5, 3)$ by $\begin{pmatrix} -2 \\ 4 \end{pmatrix}$: move $2$ left and $4$ up to get $(3, 7)$.

A translation slides every point by the same column vector, with no turning.

(Extended: a question may ask you to combine two transformations and describe the single transformation that has the same effect.)

Forces like wind are vectors, with both size and direction.

A vector has both size and direction. It can be written as a column vector, as $\overrightarrow{AB}$ (from $A$ to $B$), or in bold as $\mathbf{a}$.

• Add or subtract by working on the top and bottom numbers separately.
• Multiply by a scalar 标量 (an ordinary number) by multiplying both numbers.

Worked example. If $\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ -4 \end{pmatrix}$, then

Adding by the triangle law: draw $\mathbf{b}$ from the tip of $\mathbf{a}$, and $\mathbf{a}+\mathbf{b}$ runs from start to finish.

The magnitude 模 (length) of a vector is found with Pythagoras. For $\begin{pmatrix} x \\ y \end{pmatrix}$,

Worked example. The magnitude of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ is $\sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5$.

The magnitude of $\begin{pmatrix}3\\4\end{pmatrix}$ comes from Pythagoras on its horizontal and vertical parts.

A vector can be drawn as a directed line segment 有向线段 (an arrow). The position vector 位置向量 of a point is the vector from the origin $O$ to that point.

A key idea: the vector from $A$ to $B$ is

where $\mathbf{a}$ and $\mathbf{b}$ are the position vectors of $A$ and $B$.

To go from $A$ to $B$, travel back along $\mathbf{a}$ to $O$ then forward along $\mathbf{b}$: so $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$.

Two vectors are parallel 平行 if one is a scalar multiple of the other (for example $\overrightarrow{AB} = 2\,\overrightarrow{CD}$). Three points are collinear 共线 (in a straight line) if the vectors between them are parallel and share a point. You can express any vector in terms of two coplanar 共面 vectors.

Worked example. $O$ is the origin, with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $M$ is the midpoint of $AB$. Find $\overrightarrow{OM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

• Fully describe each transformation: a translation (a vector), a reflection (the mirror line), a rotation (centre, angle and direction), an enlargement (centre and scale factor).
• A negative scale factor turns the image upside down through the centre; a fractional one (between 0 and 1) makes it smaller.
• Add vectors tip-to-tail — add the top numbers, then the bottom numbers. The magnitude of a vector comes from Pythagoras on its components.
• $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ (end minus start). Two vectors are parallel if one is a scalar multiple of the other.