Find the image point Q' when Q(3, -1) is translated by vector pmatrix -5 \ 4 pmatrix.

Q'(-2, 3). Adding the position vector pmatrix 3 \ -1 pmatrix and translation vector pmatrix -5 \ 4 pmatrix gives pmatrix 3 + (-5) \ -1 + 4 pmatrix = pmatrix -2 \ 3 pmatrix. Thus, the coordinates of the image point are Q'(-2, 3).

Grade 10 · Core Mathematics · Lesson 19 of 21

Determining Translation Vector as a Transformation

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners express coordinates of points as column position vectors.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L1940 minutes · 3 slides

KICD learning outcome (k)

Determine translation vector as a transformation.

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

Express coordinates of points as column position vectors.
Apply translation vectors to transform 2D shapes on a Cartesian plane.
Determine translation vectors given object and image points.

Key vocabulary

Translation vector: A column vector describing the shift of a geometric figure in a given direction.; Vector $(\begin{array}{c} 4 \\ 2 \end{array})$ shifts point $A (1, 3)$ to image $A^{'} (5, 5)$ .
Congruent: Having the exact same size and shape, with identical corresponding side lengths and angle measures.; The original triangle and its translated image are congruent because their side lengths match.
Image: The new position of a geometric figure after undergoing a transformation like translation.; Point $A^{'} (4, 3)$ is the image of point $A (1, - 1)$ after translation.

Lesson preview

Slide 1 · Translating Points using Column VectorsOpening · 12 min

Figure 1: A graph showing the translation of a point P to its image P' using a column vector.

A translation shifts every point on a plane by a specified distance in a given direction without changing the shape or orientation of the object. We represent this shift using a column vector $(\begin{array}{c} a \\ b \end{array})$ , where $a$ represents the horizontal shift along the $x$ -axis and $b$ represents the vertical shift along the $y$ -axis. By adding this translation vector to the position vector of a point, we obtain the position vector of its image. Use Figure 1 to see how adding the translation vector gives the new coordinates.

A position vector represents the coordinates of a point as a column vector relative to the origin.
Adding a column vector to a position vector calculates the position vector of the translated image point.
The top number in a translation vector specifies right or left movement along the horizontal axis.
The bottom number in a translation vector specifies upward or downward movement along the vertical axis.

Worked example. Write the position vector of

P (- 2, 3)

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

. Add translation vector

T = (\begin{array}{c} 4 \\ 2 \end{array})

\vec{O P}

to find

\vec{O P^{'}}

. Calculate

\vec{O P^{'}} = (\begin{array}{c} - 2 \\ 3 \end{array}) + (\begin{array}{c} 4 \\ 2 \end{array}) = (\begin{array}{c} 2 \\ 5 \end{array})

. Answer: The position vector of $P^{'}$ is $(\begin{array}{c} 2 \\ 5 \end{array})$ , so the coordinates of $P^{'}$ are $(2, 5)$ ..

Check question: Find the image point $Q^{'}$ when $Q (3, - 1)$ is translated by vector $(\begin{array}{c} - 5 \\ 4 \end{array})$ .

$Q^{'} (- 2, 3)$
$Q^{'} (8, - 5)$
$Q^{'} (- 8, 5)$
$Q^{'} (7, - 6)$

Show answer

$Q^{'} (- 2, 3)$ . Adding the position vector $(\begin{array}{c} 3 \\ - 1 \end{array})$ and translation vector $(\begin{array}{c} - 5 \\ 4 \end{array})$ gives $(\begin{array}{c} 3 + (- 5) \\ - 1 + 4 \end{array}) = (\begin{array}{c} - 2 \\ 3 \end{array})$ . Thus, the coordinates of the image point are $Q^{'} (- 2, 3)$ .

Slide 2 · Transforming Shapes by Translation VectorsModel · 14 min

Figure 2: A graph demonstrating how a shape is translated by moving each vertex by the same vector.

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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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