Video: Describing Translations of Shapes with Vectors

Shape A has been translated to shape B and then to shape C. Write a vector to represent the translation from shape A to shape B. Write a vector to represent the translation from shape B to shape C. Write a vector to represent the translation from shape C to shape A.

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Video Transcript

Shape A has been translated to shape B and then to shape C. Write a vector to represent the translation from shape A to shape B. Write a vector to represent the translation from shape B to shape C. Write a vector to represent the translation from shape C to shape A.

We know that a translation is simply a shift or movement of an object from one position to another. In each part of this question, we were asked to describe these translations using vectors. We recall that the convention when writing this vector is to write the number of units right, first of all, followed by the number of units up. Let’s consider, then, the translation from shape A to shape B first of all. And we’ll use a pair of corresponding corners or vertices on the two shapes.

Considering the horizontal movement, first of all, and counting the squares, we can see that shape A has been translated five units to the left. Moving five units to the left is the same as moving negative five units to the right. So, the first number in our column vector to describe the horizontal translation is negative five. Vertically, we then see that shape A has been translated two units down. And a translation two units down is equivalent to a translation of negative two units up. So, we can fill in the second number in our column vector. The vector that represents the translation from shape A to shape B then is the column vector negative five, negative two. And the negatives indicate that the translation is to the left and down.

Now, let’s consider the vector that represents the translation from shape B to shape C. We can use a different pair of corresponding vertices this time if we wish. We can see that the general direction of movement or translation from shape B to shape C is to the right and then down. This means that we’re expecting a positive number for the first number in our column vector and a negative number for the second.

Looking at the horizontal movement, first of all, we can see that this vertex moves three units to the right. So, we can express this using positive three. Looking vertically, we can see that the shape is translated four units down. So, we express this as negative four. Remember, translation of negative four units up is equivalent to a translation of four units down. So, the vector that represents the translation from shape B to shape C is the column vector three, negative four.

Finally, we need to write the vector that represents the translation from shape C to shape A. And we’ll pick a different pair of corresponding vertices to use again this time. The direction of movement here is to the right and up. So, both values in our column vector will be positive. Looking horizontally, first of all, we see that shape C is translated two units to the right. So, the first value in our column vector is two. And then, looking vertically, we see that shape C is translated six units up. So, the vector that represents the translation from shape C to shape A is the column vector two, six.

Now, we can observe something interesting here. Which is that we could find the vector that would describe the translation from shape A to shape C by adding together the vector that describes the translation from A to B and then the vector that describes the translation from B to C. If we did this, then adding the component parts of our two vectors together, we’d find that the vector that represents the translation from A to C is negative two, negative six. Which we can see is the exact negative of the vector that represents the translation from C to A.

The reason for this is that these two translations are at the exact same distances but in the opposite direction. To go from shape A to shape C, we’re translating the object left and down. Whereas to go from shape C to shape A, we’re translating right and up. We have our three answers for this problem though. The three vectors are negative five, negative two; three, negative four; and two, six.

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