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Translations and Vectors

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Translations

Translations are often informally called “slides”. Why is this?

Watch This

http://www.youtube.com/watch?v=4g5uRFzIYBM Brightstorm: Translations

Guidance

A translation is one example of a rigid transformation . A translation moves each point in a shape a specified distance in a specified direction as defined by a vector. Below, the parallelogram has been translated along vector \vec{v}  to create a new parallelogram (the image).

Keep in mind that the location of vector \vec{v}   does not matter . Vectors have a direction and a magnitude (a length), and simply tell you how to move points. Vector  \vec{v} essentially tells you that all points move three units to the right and one unit up

The lines that connect corresponding points will all be parallel to vector \vec{v} .

With the grid in the background, you can see that the slope of vector  \vec{v} and each line is \frac{1}{3} . This should make sense because each point in the original parallelogram was moved 3 units to the right and 1 unit up to create its corresponding point in the image. 

One way to think about translations is that they move points a specified distance along lines parallel to a given line.  In this case, all points were moved a distance of \sqrt{10}  (found using the Pythagorean Theorem) along lines parallel to vector \vec{v} .

If you are performing a translation of a shape not on a grid, the vector becomes crucial. You can no longer say “move one unit up and three units to the right” because without a grid there are no units. Below is a translation of another quadrilateral without a grid in the background.

Notice that lines parallel to vector  \vec{v} have been drawn through each of the original points. Vector  \vec{v} has been copied onto each of those lines at the points that define the original quadrilateral. The ends of the vectors define the points on the image.

Example A

Is the following transformation a translation?

Solution: One way to check if a transformation is a translation is to look at how each point moves to create its image. If all points move in the same way, then it is a translation.

Point to Image Point

Description of Motion

A  to A^\prime

4 to the right and 1 up

B  to B^\prime

4 to the right and 1 up

C  to C^\prime

4 to the right and 2 up

D  to D^\prime

4 to the right and 1 up

Because  C to  C^\prime is different, this is not a translation.

Example B

Describe the vector that defined the translation below.

Solution: The vector moved each point three units to the right and three units up.

Example C

Perform the translation defined by vector  \vec{u} on the quadrilateral below.

Solution: With the grid in the background, you can see that vector  \vec{u} tells you to move each point 2 units up and 1 unit to the left. Here is the translation:

Concept Problem Revisited

A translation is informally called a slide, because it essentially slides a shape to a new position. The orientation of the points does not change.

Vocabulary

A translation is a rigid transformation that moves each point in a shape a specified distance in a specified direction as defined by a vector.

Guided Practice

1. Describe the vector that defined the translation below.

2. Perform the translation defined by vector  \vec{t} on the triangle below.

3. Is the following transformation a translation?

Answers:

1. The vector moved each point 4 units to the right and 2 units down.

2. 

3. Yes, each point moves one unit to the right and two units up.

Practice

1. Is a translation a rigid transformation? Explain.

2. What role does a vector play in a translation?

3. How are parallel lines relevant to translations?

4. How can you tell if a transformation is a translation?

Describe the vector that defined each of the following translations.

5.

6.

7.

8.

Perform the translation defined by vector  \vec{t} on the polygons below.

9.

10.

11.

12.

Are the following transformations translations? Explain.

13.

14.

15.

Image Attributions

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