The translation of vectors in the plane is an application to a point by means of a vector called the translation vector, resulting in a point called the homologous point.
In other words, the translation of vectors is to move a point by adding the coordinates of the translation vector and obtaining another point called the homologous point.
It is called point modification or transformation since the translation is applied to a point in the plane through a vector.
Vector translation procedure
To do a translation of vectors we need:
- A point in the plane.
- A vector that will dictate the translation of the point.
- Do the translation: obtain the homologous point.
Once the first two steps above have been applied, we will obtain a point called the homologous point (step 3). This point is the result of making the translation of the initial point. So, to be able to translate a vector, it is essential to have a point and a vector, otherwise, we will not be able to do the translation.
First step: Have a point
We assume that we have an initial point called P and that it has x and y coordinates. We add the subscript p to emphasize that these coordinates belong to the point P.
Second step: Have a vector
Also, we assume that we have a vector called v that is between points A and B.
Third step: Make the translation
Once we have controlled the point and the vector from which we will apply the translation, we only need to calculate the homologous point. The homologous point is calculated by adding the coordinates of the vector and the initial point. In our case, we would have to write the coordinates of the initial point P and add the coordinates of the vector v to it.
The pink point corresponds to the point homologous to point P. To obtain the coordinates of the homologous point we have to add the coordinates of the vector v to the coordinates of the initial vector P.
We can see in the graph how the homologous point is found with the translation of the point P by means of the vector v. Through the addition of the coordinates we are moving the point P in another situation in the plane that meets the coordinates of the vector v.
Translation of vectors applied to figures
In the same way that the translation has been applied to a specific point, it can also be applied to a set of points. If this set of points is connected by segments and is closed, we will speak of a figure. Then, the translation of geometric figures in the plane can be applied.
Vector translation example
Apply the translation to the next point using the following vector: