Copy of 14.7d The Inverse of a Translations
What is the inverse of a translation, or composition of translations?
1. Click on the checkbox to begin, and create a translation for triangle ABC by dragging the brown vector. Note that a or b or both can be negative.
2. Click on the second checkbox and drag the red vector to create the translation that maps triangle A'B'C' back to its pre-image, ABC. What are the values for c and d?
3. Reset the vectors to their original positions, (0, 1), and click on the third checkbox.
4. Drag the brown vector to create translation T. Drag the red vector for T-squared. Note that T-squared means the translation T applied twice, first to the pre-image ABC, and then to the image of ABC under T. What are the values for c and d?
5. Drag the orange vector to map the second image back to the pre-image ABC. What are the values of f and g?
The inverse of a translation adds the opposite values to each of the coordinates so that the composition of the translation, T, composed with T-inverse, results in the identity. Verify that this is commutative.
As you know, the composition of a translation is the sum of the translated coordinates, so that the inverse of the composition is the sum of the opposites of the coordinates, resulting in the identity.