 Translation by a Vector

Lesson

Translation by a vector refers to the movement of one or more points of a space in a particular direction by a specified amount.

A vector is often represented geometrically as an arrow with a particular direction and length. We can slide the arrow to any location in the space, without changing its direction or length, and it is considered to be the same vector.

So, to translate a point in the space by a given vector, we place the tail of the arrow at the point. The translated point is then at the head of the arrow. This process is called vector addition. It is illustrated in the diagram below. To accomplish the same result algebraically, we represent the points by their coordinates. The translation vector is also represented by its coordinates. These are the distances by which the points are to be moved in the directions of each of the coordinate axes.

For example, the coordinate pair $\left(a,b\right)$(a,b) represents a translation vector where points are to be moved $a$a units horizontally and $b$b units vertically.  When applied to any point $\left(x,y\right)$(x,y), the result is the translated point $\left(x+a,y+b\right)$(x+a,y+b).

Vector addition to translate points is also valid in spaces of three (or more) dimensions. For example, an arbitrary point $\left(x,y,z\right)$(x,y,z) in ordinary three-dimensional space is translated by the vector $\left(a,b,c\right)$(a,b,c) to the new point $\left(x+a,y+b,z+c\right)$(x+a,y+b,z+c)

Example 1

The points $(1,-2)$(1,2), $(2.5,3)$(2.5,3) and  $(-1,0)$(1,0) in the Cartesian plane represent the corners of a triangular shape.  A copy of this shape, such that all three corners have only positive coordinates, can be obtained by many different translations of the original triangle. One such translation is by the vector $\left(2,3\right)$(2,3), Find the coordinates of the corners of the copy under this translation.

We apply the translation to each of the corner points by vector addition. Thus, the corners of the copy are at $\left(1+2,-2+3\right)=(3,1)$(1+2,2+3)=(3,1), $\left(2.5+2,3+3\right)=(5.5,6)$(2.5+2,3+3)=(5.5,6) and $\left(-1+2,0+3\right)=(1,3)$(1+2,0+3)=(1,3).

Example 2

What translation would be needed to bring the point $(-1,0)$(1,0) in Example $1$1 to the origin of the coordinate axes? Find the new locations of the corners under this translation.

We need to add the vector $(1,0)$(1,0) since $(-1,0)+(1,0)=(0,0)$(1,0)+(1,0)=(0,0). The other two corners are at $\left(2.5+1,3+0\right)=(3.5,3)$(2.5+1,3+0)=(3.5,3) and $\left(1+1,-2+0\right)=(2,-2)$(1+1,2+0)=(2,2).