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Vectors and Transformations - Translations, Rotations, and Reflections

Grade 12IGCSE

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

🔑Concepts

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Column Vectors: Represented as (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 (axis of symmetry). Every point and its image are equidistant from this line.

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Rotation: Turning a shape around a fixed center point by a specific angle and direction (clockwise or anti-clockwise).

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Invariant Points: Points that do not change position after a transformation (e.g., the center of rotation).

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Transformation Matrices: Using 2×22 \times 2 matrices to represent linear transformations like reflection and rotation centered at the origin (0,0)(0,0).

📐Formulae

Magnitude of a vector: ∣v∣=x2+y2|\mathbf{v}| = \sqrt{x^2 + y^2}

Translation mapping: P′=P+(ab)P' = P + \begin{pmatrix} a \\ b \end{pmatrix}

Reflection in x-axis: (100−1)\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

Reflection in y-axis: (−1001)\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}

Reflection in y=xy = x: (0110)\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Rotation 90∘90^\circ Anti-clockwise about (0,0)(0,0): (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

Rotation 180∘180^\circ about (0,0)(0,0): (−100−1)\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}

Rotation 270∘270^\circ Anti-clockwise (or 90∘90^\circ Clockwise): (01−10)\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}

💡Examples

Problem 1:

A triangle with vertices A(1,2)A(1, 2), B(3,2)B(3, 2), and C(1,4)C(1, 4) is translated by the vector v=(−23)\mathbf{v} = \begin{pmatrix} -2 \\ 3 \end{pmatrix}. Find the coordinates of the image A′B′C′A'B'C'.

Solution:

A′=(1−2,2+3)=(−1,5)A' = (1-2, 2+3) = (-1, 5), B′=(3−2,2+3)=(1,5)B' = (3-2, 2+3) = (1, 5), C′=(1−2,4+3)=(−1,7)C' = (1-2, 4+3) = (-1, 7)

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 of the point.

Problem 2:

Reflect the point P(4,−2)P(4, -2) in the line y=xy = x.

Solution:

P′=(−2,4)P' = (-2, 4)

Explanation:

When reflecting in the line y=xy=x, the xx and yy coordinates are swapped. Using matrix multiplication: (0110)(4−2)=(−24)\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}.

Problem 3:

Rotate the point Q(3,1)Q(3, 1) 90∘90^\circ clockwise about the origin (0,0)(0,0).

Solution:

Q′=(1,−3)Q' = (1, -3)

Explanation:

A 90∘90^\circ clockwise rotation is equivalent to a 270∘270^\circ anti-clockwise rotation. Applying the matrix (01−10)(31)\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 3 \\ 1 \end{pmatrix} results in (1,−3)(1, -3).