Transformations and Vectors

The One Thing

Transformations describe how shapes change position, orientation or size according to precise defining information. Vectors provide an efficient coordinate method for representing movement and proving geometric relationships involving direction, length, ratios, parallelism and collinearity.

Definitions and Results

• Object and image: The original shape is the object; the shape after a transformation is the image.
• Reflection: A flip in a mirror line. Each point and its image are the same perpendicular distance from the mirror line.

- In the $x$-axis: $(x,y)\mapsto (x,-y)$. - In the $y$-axis: $(x,y)\mapsto (-x,y)$. - In the line $y=x$: $(x,y)\mapsto (y,x)$. - In the line $y=-x$: $(x,y)\mapsto (-y,-x)$.

• Rotation: A turn about a fixed centre through a specified angle and direction, usually clockwise or anticlockwise.

- $90^{\circ }$ anticlockwise about the origin: $(x,y)\mapsto (-y,x)$. - $90^{\circ }$ clockwise about the origin: $(x,y)\mapsto (y,-x)$. - $180^{\circ }$ about the origin: $(x,y)\mapsto (-x,-y)$.

• Translation: A slide in which every point moves the same distance in the same direction. Translation by $(a,b)$ maps

$$(x,y)\mapsto (x+a,y+b).$$

• Enlargement: A transformation that changes size using a scale factor and a centre of enlargement.

- With centre at the origin and scale factor $k$: $$(x,y)\mapsto (kx,ky).$$ - If $k>1$, the image is enlarged. - If $0<k<1$, the image is reduced. - If $k<0$, the image lies on the opposite side of the centre and its distances from the centre are multiplied by $|k|$. - For a general centre, the image point lies on the line from the centre through the original point, at a distance multiplied by the scale factor.

• Invariant: A property or point that does not change under a transformation. Reflections, rotations and translations preserve lengths and angles. Enlargements preserve angles but multiply all lengths by the absolute value of the scale factor.
• Column vector: A vertically written vector such as

$$\left(\begin{array}{c}a\\ b\end{array}\right),$$ representing a movement of $a$ units horizontally and $b$ units vertically.

• Magnitude: The length of a vector. For $(a,b)$,

$$|(a,b)|=\sqrt{a^2+b^2}.$$

• Position vector: A vector from the origin to a point, commonly written as the point’s coordinate column vector.
• Equal vectors: Vectors with the same magnitude and direction, even when they begin at different points.
• Vector addition:

$$\left(\begin{array}{c}a\\ b\end{array}\right)+\left(\begin{array}{c}c\\ d\end{array}\right)=\left(\begin{array}{c}a+c\\ b+d\end{array}\right).$$

• Scalar multiplication:

$$k\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}ka\\ kb\end{array}\right).$$ A negative scalar reverses the direction as well as changing the magnitude.

• Vector from $A$ to $B$:

$$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}.$$

• Reverse vector:

$$\overrightarrow{BA}=-\overrightarrow{AB}.$$

• Connected vector path: For points $A$, $B$ and $C$,

$$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}.$$

• Parallel vectors: Vectors that are scalar multiples of one another.
• Collinearity: Points are collinear if the vectors joining them are parallel, or if one point can be written as a position-vector combination on the line through the other two.
• Parallelogram: A quadrilateral is a parallelogram if opposite sides are equal and parallel, or if its diagonals have the same midpoint.
• Transformation matrix: A matrix can represent a coordinate transformation. A $90^{\circ }$ anticlockwise rotation about the origin has matrix

$$\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right).$$

• Determinant and area: The absolute value of the determinant of a transformation matrix gives the area scale factor for a linear transformation.

Worked Methods

Applying a reflection

• Identify the mirror line.
• Apply the appropriate coordinate rule to every vertex.
• Check that each point and its image are the same perpendicular distance from the mirror line.

For reflection in the $x$-axis: $$(x,y)\mapsto (x,-y).$$

For reflection in the $y$-axis: $$(x,y)\mapsto (-x,y).$$

For reflection in $y=x$: $$(x,y)\mapsto (y,x).$$

For reflection in $y=-x$: $$(x,y)\mapsto (-y,-x).$$

Applying a rotation about the origin

• Identify the angle and direction.
• Apply the relevant rule to every vertex.
• Check that distances from the centre and the shape’s lengths are unchanged.

For $90^{\circ }$ anticlockwise: $$(x,y)\mapsto (-y,x),\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}-y\\ x\end{array}\right).$$

For $90^{\circ }$ clockwise: $$(x,y)\mapsto (y,-x).$$

For $180^{\circ }$: $$(x,y)\mapsto (-x,-y).$$

Applying a translation

• Identify the translation vector $(a,b)$.
• Add $a$ to every $x$-coordinate.
• Add $b$ to every $y$-coordinate.
• Check that all corresponding points have moved by the same vector.

The rule is $$(x,y)\mapsto (x+a,y+b).$$

Applying an enlargement

• Identify the centre of enlargement and scale factor $k$.
• If the centre is the origin, multiply both coordinates by $k$:

$$(x,y)\mapsto (kx,ky).$$

• If the centre is general, draw a line from the centre through each original point.
• Place the image point on the same line at a distance multiplied by $k$.
• Interpret the scale factor:

- $k>1$: enlargement; - $0<k<1$: reduction; - $k<0$: image on the opposite side of the centre.

Finding a vector between two points

• Write the position vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$.
• Subtract the first from the second:

$$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}.$$

• Reverse the vector when required:

$$\overrightarrow{BA}=-\overrightarrow{AB}.$$

Calculating a vector magnitude

For $$\mathbf{v}=\left(\begin{array}{c}a\\ b\end{array}\right),$$ use $$|\mathbf{v}|=\sqrt{a^2+b^2}.$$

Adding vectors and following a path

• Write each vector as a column vector.
• Add corresponding components:

$$\left(\begin{array}{c}a\\ b\end{array}\right)+\left(\begin{array}{c}c\\ d\end{array}\right)=\left(\begin{array}{c}a+c\\ b+d\end{array}\right).$$

• For a connected path through $A$, $B$ and $C$, use

$$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}.$$

Proving parallelism or collinearity

• Calculate the relevant joining vectors.
• Compare their components.
• Show that one vector is a scalar multiple of the other.
• Conclude that the vectors are parallel.
• If the vectors join points on the same line, conclude that the points are collinear.

Alternatively, express one point as a position-vector combination on the line through the other two.

Proving that a quadrilateral is a parallelogram

Use either method:

• Calculate opposite side vectors and show that each pair is equal and parallel; or
• Calculate the midpoints of the diagonals and show that they are the same.

Dividing a line segment in a ratio

• Write the position vectors of the two endpoints.
• Use the given ratio to form a weighted average of those position vectors.
• The resulting position vector locates the point dividing the segment in that ratio.

Using a transformation matrix

• Write the point as a coordinate column vector.
• Multiply it by the transformation matrix.
• Interpret the resulting column vector as the image coordinates.
• Use the determinant to determine the area scale factor; the relevant factor is the absolute value of the determinant.

Where It Goes Wrong

• Applying a transformation rule without first identifying the defining information: the mirror line, centre and angle, translation vector, or centre and scale factor.
• Reversing the direction of a rotation, particularly confusing $90^{\circ }$ clockwise $(x,y)\mapsto (y,-x)$ with $90^{\circ }$ anticlockwise $(x,y)\mapsto (-y,x)$.
• Forgetting that a translation moves every point by the same vector, or adding the vector components inconsistently.
• Treating an enlargement with a negative scale factor as an ordinary positive enlargement; a negative factor places the image on the opposite side of the centre.
• Confusing $\overrightarrow{AB}$ with $\overrightarrow{BA}$; the reverse vector is $\overrightarrow{BA}=-\overrightarrow{AB}$.
• Omitting the conditions in vector proofs: parallel vectors must be scalar multiples, and a parallelogram can be established through equal parallel opposite sides or diagonals with the same midpoint.

What Gets Asked

• Determine the image of a shape under a reflection in the $x$-axis, $y$-axis, $y=x$, or $y=-x$.
• Find the image of a point or shape after a $90^{\circ }$ clockwise, $90^{\circ }$ anticlockwise, or $180^{\circ }$ rotation about the origin.
• Apply a translation given by a column vector.
• Apply an enlargement from the origin or from a general centre, including fractional and negative scale factors.
• Identify a transformation from its coordinate rule, matrix, or effect on a shape.
• Calculate a vector between two points using

$$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}.$$

• Find the magnitude of a vector using

$$\sqrt{a^2+b^2}.$$

• Add vectors, apply scalar multiplication, or use a connected path such as

$$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}.$$

• Prove that vectors are parallel or that points are collinear.
• Find a point dividing a line segment in a given ratio using weighted position vectors.
• Prove that a quadrilateral is a parallelogram using opposite sides or diagonal midpoints.
• Use a transformation matrix to calculate image coordinates or determine the area scale factor from its determinant.