Grade 10 · Core Mathematics · Lesson 20 of 21

Determining Translation Vectors as Transformations

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners determine the translation vector mapping an object point to an image point.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L2040 minutes · 3 slides

KICD learning outcome (k)

Determine translation vector as a transformation.

By the end of this lesson, learners should be able to

Determine the translation vector mapping an object point to an image point.
Calculate original object coordinates given an image point and translation vector.
Translate geometric shapes on the Cartesian plane using column vectors.

Key vocabulary

Translation vector: A column vector describing the magnitude and direction of horizontal and vertical shift.; The vector $(\begin{array}{c} 4 \\ 7 \end{array})$ maps point $A (3, 2)$ onto $A^{'} (7, 9)$ .
Image: The new position of a point or shape resulting from a geometric transformation.; Point $B^{'} (4, 7)$ is the image of $B (- 1, 10)$ under a translation.
Transformation: A mathematical operation that changes the position, size, or orientation of a shape.; Translation is a transformation mapping $A (- 2, 1)$ to $A^{'} (3, 8)$ .

Lesson preview

Slide 1 · Determining Translation Vectors and Image PointsModel · 12 min

Figure 1: A coordinate plane shows the translation vector mapping point A to A' and point B to B'.

A translation is a transformation that shifts every point on a plane by the same distance in a specified direction. The translation vector $T = (\begin{array}{c} a \\ b \end{array})$ gives the horizontal shift $a$ and vertical shift $b$ . When a point $P (x, y)$ is mapped onto $P^{'} (x^{'}, y^{'})$ , the vector equation $\vec{O P^{'}} = \vec{O P} + T$ allows us to determine $T = \vec{O P^{'}} - \vec{O P}$ . We then use this same vector to translate any other point.

The translation vector $T$ is calculated as the image position vector minus the object position vector $T = \vec{O P^{'}} - \vec{O P}$ .
Horizontal displacement $a$ is the top component, and vertical displacement $b$ is the bottom component of $(\begin{array}{c} a \\ b \end{array})$ .
Adding the translation vector $T$ to any point's position vector gives the exact position vector of its image.
Every point of a translated shape moves along parallel paths of equal length determined by vector $T$ .

Worked example. Write position vectors

\vec{O A} = (\begin{array}{c} 3 \\ 2 \end{array})

and

\vec{O A^{'}} = (\begin{array}{c} 7 \\ 9 \end{array})

. Calculate translation vector

T = \vec{O A^{'}} - \vec{O A} = (\begin{array}{c} 7 \\ 9 \end{array}) - (\begin{array}{c} 3 \\ 2 \end{array}) = (\begin{array}{c} 4 \\ 7 \end{array})

. Apply vector

T

B

\vec{O B^{'}} = \vec{O B} + T = (\begin{array}{c} 8 \\ 11 \end{array}) + (\begin{array}{c} 4 \\ 7 \end{array}) = (\begin{array}{c} 12 \\ 18 \end{array})

. Answer: The translation vector is $(\begin{array}{c} 4 \\ 7 \end{array})$ and the image point is $B^{'} (12, 18)$ ..

Check question: A translation maps point $M (- 2, 5)$ onto $M^{'} (1, 8)$ . What is the image of $N (4, - 1)$ under this same translation?

$N^{'} (3, 12)$
$N^{'} (1, - 4)$
$N^{'} (3, 2)$
$N^{'} (7, 2)$

Show answer

$N^{'} (7, 2)$ . First, find the translation vector $T$ by subtracting the object position vector from the image position vector: $T = (\begin{array}{c} 1 \\ 8 \end{array}) - (\begin{array}{c} - 2 \\ 5 \end{array}) = (\begin{array}{c} 1 - (- 2) \\ 8 - 5 \end{array}) = (\begin{array}{c} 3 \\ 3 \end{array})$ . To find the image of $N$ , add $T$ to $\vec{O N}$ : $\vec{O N^{'}} = (\begin{array}{c} 4 \\ - 1 \end{array}) + (\begin{array}{c} 3 \\ 3 \end{array}) = (\begin{array}{c} 7 \\ 2 \end{array})$ . Therefore, the image is $N^{'} (7, 2)$ .

Slide 2 · Finding Original Object Points and Translating ShapesPractice · 14 min

Figure 2: A coordinate grid displays an image point reversed to its object point by subtracting the translation vector.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

All 3 slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 20 lessons in this sub-strand

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