Translations and Vectors
A translation is a key mathematical transformation that slides a shape from one place to another without turning or resizing it. We use special instructions called vectors to describe this movement precisely — a skill used everywhere from video game design to sports coaching.
Understanding Translations and Column Vectors
What is a Translation?
A translation is a “slide”. Every point on the shape moves the exact same distance in the exact same direction. The shape’s size and orientation do not change.
Using a Column Vector
We use a column vector to give the precise instructions for the slide.
Example: The vector $\begin{pmatrix} 4 \\ -2 \end{pmatrix}$ means move 4 units right and 2 units down.
Worked Examples
Example 1: Translating a Shape
Translate triangle PQR by the vector $\begin{pmatrix} -4 \\ -3 \end{pmatrix}$.
Move every point 4 left and 3 down.
Example 2: Finding the Vector
Describe the translation from shape A to shape B using a column vector.
- Pick a corresponding point on both shapes, e.g., the bottom-left corner.
- Horizontal change ($x$): From $x=1$ to $x=4$ → $+3$.
- Vertical change ($y$): From $y=1$ to $y=3$ → $+2$.
Answer: $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$
Tutor Insights
🤔 Common Misunderstandings
- Mixing up $x$ and $y$. Putting the vertical movement on top and horizontal on the bottom. Remember: $x$ is across, $y$ is up/down.
- Incorrect signs. Forgetting that moving left means a negative $x$-value and moving down means a negative $y$-value.
📝 Common Exam Mistakes
- Flipping the vector components in the final answer.
- Miscounting squares on the grid when drawing or describing a translation.
- Not translating all vertices, leading to an incorrectly drawn image.
Practice Questions
- Point P is at $(3, 2)$. It is translated by $\begin{pmatrix} 2 \\ 4 \end{pmatrix}$. What are the coordinates of its image, P’?
- Triangle ABC has vertices $A(1,1)$, $B(4,1)$, $C(2,3)$. It is translated by $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$. What are the coordinates of A’B’C’?
- An object is at $(5, 6)$ and its image after a translation is at $(1, 3)$. Write down the column vector for this translation.
- A shape is translated by $\begin{pmatrix} 2 \\ -5 \end{pmatrix}$. If one point on the image is at $(0, 0)$, what were the coordinates of the corresponding point on the original object?
Show Answers
- Working: New $x = 3+2 = 5$. New $y = 2+4 = 6$.
Answer: $(5, 6)$. - Working: $A(1,1) \to (-2,3)$. $B(4,1) \to (1,3)$. $C(2,3) \to (-1,5)$.
Answer: A'(−2, 3), B'(1, 3), C'(−1, 5). - Working: $x$-movement $= 1-5 = -4$. $y$-movement $= 3-6 = -3$.
Answer: $\begin{pmatrix} -4 \\ -3 \end{pmatrix}$. - Working: Apply the reverse vector $\begin{pmatrix} -2 \\ 5 \end{pmatrix}$ to $(0,0)$.
Answer: $(−2, 5)$.
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