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Vectors and Transformations - Translations, Rotations, and Reflections

Grade 12A Level

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

🔑Concepts

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A translation moves every point of a shape the same distance in the same direction. It is described by a column vector (xy)\begin{pmatrix} x \\ y \end{pmatrix}, where xx represents horizontal movement (right is positive) and yy represents vertical movement (up is positive).

A triangle translated by vector (2, 1) on a coordinate plane.
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Reflections flip a shape over a mirror line. Points on the mirror line remain invariant. Every point of the image is at the same perpendicular distance from the mirror line as the original point but on the opposite side.

Reflection of a triangle across the line y=x.
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Rotations turn a shape around a fixed point called the center of rotation. A rotation is defined by the center, the angle, and the direction (clockwise or anti-clockwise). For origin rotations, specific matrices are used.

Rectangle rotated 90 degrees anti-clockwise about the origin.
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An invariant point is a point that does not change its position under a transformation. For reflections, all points on the mirror line are invariant. For rotations, only the center of rotation is invariant.

📐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).

Problem 4:

Reflect the square with vertices S(1,1)S(1, 1), T(2,1)T(2, 1), U(2,2)U(2, 2), and V(1,2)V(1, 2) in the yy-axis. Find the new coordinates.

Square reflected across the y-axis.

Solution:

The transformation for reflection in the yy-axis is (x,y)→(−x,y)(x, y) \rightarrow (-x, y). Applying this to each vertex: S′(−1,1)S'( -1, 1 ) T′(−2,1)T'( -2, 1 ) U′(−2,2)U'( -2, 2 ) V′(−1,2)V'( -1, 2 ) The image is a square on the left side of the yy-axis.

Explanation:

Reflection in the yy-axis negates the x-coordinate while the y-coordinate remains unchanged. The distance of each point from the yy-axis remains the same.

Problem 5:

A triangle with vertices L(0,0)L(0, 0), M(2,0)M(2, 0), and N(0,1)N(0, 1) is rotated 180∘180^\circ about the origin. Determine the coordinates of the image L′M′N′L'M'N'.

Triangle rotated 180 degrees about the origin.

Solution:

The transformation for a 180∘180^\circ rotation about the origin is (x,y)→(−x,−y)(x, y) \rightarrow (-x, -y). L′(0,0)L'( 0, 0 ) M′(−2,0)M'( -2, 0 ) N′(0,−1)N'( 0, -1 ) The point LL is at the origin (center of rotation), so it is an invariant point.

Explanation:

In a 180∘180^\circ rotation about (0,0)(0,0), both the xx and yy coordinates change sign. This is equivalent to reflecting the shape in both the xx and yy axes.

Translations, Rotations, and Reflections Grade 12 Notes & Examples