6.6 Translation by a Vector

6.6 Translation by a Vector

A translation vector is a vector that describes the movement of a point, a body, a system from one position to another without rotation or deformation. The translated figure is called the image of original figure. The image is congruent to the original figure.

To translate a figure with translation vector in a coordinate plane, we change the coordinates of its points. If $\underline{v} = a\underline{i} + b\underline{j}$ is the translation vector and $a, b$ are constants, then we can use the guidelines below:

Slide up $b$ units
$y \rightarrow y + b$
$\uparrow$
Slide to the left $a$ unit $\leftarrow$ $\rightarrow$ Slide to the right $a$ unit
$x \rightarrow x - a$ $x \rightarrow x + a$
$\downarrow$
Slide down $b$ units
$y \rightarrow y - b$

Let us translate point $P(2, 1)$ by translation vectors $2\underline{j}, 3\underline{i}, -3\underline{j}$ and $-5\underline{i}$.

$P_1 = (2\underline{i} + \underline{j}) + (0\underline{i} + 2\underline{j})$

$= 2\underline{i} + 3\underline{j}$

$P_2 = (2\underline{i} + \underline{j}) + (3\underline{i} + 0\underline{j})$

$= 5\underline{i} + \underline{j}$

$P_3 = (2\underline{i} + \underline{j}) + (0\underline{i} - 3\underline{j})$

$= 2\underline{i} - 2\underline{j}$

$P_4 = (2\underline{i} + \underline{j}) + (-5\underline{i} + 0\underline{j})$

$= -3\underline{i} + \underline{j}$

[Figure 6.13]

Example 9

Plot $A(-4, 2), B(-1, 2)$ and $C(-3, 4)$ to form a triangle $ABC$. Also translate $\Delta ABC$ to $\Delta A'B'C'$ by the translation vector $7\underline{i} + 2\underline{j}$.

Solution

New vertex $A' = (-4\underline{i} + 2\underline{j}) + (7\underline{i} + 2\underline{j})$

$= 3\underline{i} + 4\underline{j} = (3, 4)$

New vertex $B' = (-\underline{i} + 2\underline{j}) + (7\underline{i} + 2\underline{j})$

$= 6\underline{i} + 4\underline{j} = (6, 4)$

New vertex $C' = (-3\underline{i} + 4\underline{j}) + (7\underline{i} + 2\underline{j})$

$= 4\underline{i} + 6\underline{j} = (4, 6)$

[Figure 6.14]