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Vectors and Transformations - Reflection, Rotation, Translation, and Enlargement

Grade 9IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Vectors: Represented as column vectors (xy)\begin{pmatrix} x \\ y \end{pmatrix}, where xx is the horizontal displacement and yy is the vertical displacement.

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Translation: Moving a shape without rotating or resizing it. Defined by a translation vector.

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Reflection: Flipping a shape over a mirror line (e.g., x=kx=k, y=ky=k, y=xy=x, or y=−xy=-x). Every point and its image are equidistant from the mirror line.

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Rotation: Turning a shape around a fixed point (center of rotation) by a specific angle and direction (clockwise or anticlockwise).

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Enlargement: Changing the size of a shape from a center of enlargement using a scale factor kk. If k>1k > 1, the shape grows; if 0<k<10 < k < 1, it shrinks; if kk is negative, the shape is inverted.

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Invariance: Points that do not move under a transformation are called invariant points.

📐Formulae

Vector Addition: (ab)+(cd)=(a+cb+d)\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}

Scalar Multiplication: k(xy)=(kxky)k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}

Magnitude of a vector v⃗=(xy)\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}: ∣v⃗∣=x2+y2|\vec{v}| = \sqrt{x^2 + y^2}

Enlargement Area Scale Factor: Area of Image=k2×Area of Object\text{Area of Image} = k^2 \times \text{Area of Object}

Reflection in y=xy=x: (x,y)→(y,x)(x, y) \rightarrow (y, x)

Reflection in y=−xy=-x: (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x)

💡Examples

Problem 1:

Translate the point A(2,−3)A(2, -3) by the vector v⃗=(−45)\vec{v} = \begin{pmatrix} -4 \\ 5 \end{pmatrix}. Find the coordinates of the image A′A'.

Solution:

A′=(2+(−4),−3+5)=(−2,2)A' = (2 + (-4), -3 + 5) = (-2, 2)

Explanation:

To translate a point, add the xx-component of the vector to the xx-coordinate and the yy-component of the vector to the yy-coordinate.

Problem 2:

A triangle with vertices P(1,1)P(1, 1), Q(3,1)Q(3, 1), and R(1,4)R(1, 4) is reflected in the line y=xy = x. What are the new coordinates?

Solution:

P′(1,1),Q′(1,3),R′(4,1)P'(1, 1), Q'(1, 3), R'(4, 1)

Explanation:

When reflecting in the line y=xy=x, the xx and yy coordinates of each point are swapped.

Problem 3:

A square is enlarged by a scale factor of 33 from the center (0,0)(0,0). If the original area is 5 cm25 \text{ cm}^2, what is the area of the enlarged square?

Solution:

New Area=5×32=5×9=45 cm2\text{New Area} = 5 \times 3^2 = 5 \times 9 = 45 \text{ cm}^2

Explanation:

The area of an enlarged shape increases by the square of the scale factor (k2k^2).

Problem 4:

Rotate the point (2,5)(2, 5) 90∘90^\circ anticlockwise about the origin (0,0)(0,0).

Solution:

(−5,2)(-5, 2)

Explanation:

For a 90∘90^\circ anticlockwise rotation about the origin, the mapping is (x,y)→(−y,x)(x, y) \rightarrow (-y, x).