Translation Equations

Usually when we are talking about vectors, we talk about one of the standard planes. In 2-space, we generally talk about the $xy$-plane, $xz$-plane, or $yz$-plane. In 3-space we usually talk about the $xyz$-plane. However, it is important to recognize the the position of vectors are relative to the plane they are in.

# Translation of Axes

Let's say that we have some new coordinate system that is the $x'y'$-plane. The diagram below constructs a possible $x'y'$-plane on top of the standard $xy$-plane:

We can analyze the position of the new coordinate system as follows in this diagram with relation to some vector $\vec{u} = (u_1, u_2)$. Note that the position of $\vec{u}$ is different depending on it being placed in the $xy$-plane or the $x'y'$-plane.

We should notice that the origin in the $xy$-plane is $O = (0,0)$, and that the origin in the $x'y'$-plane is $O' = (k, l)$. Hence we can derive translation equations for equivalency.

(1)
\begin{align} x' = x - k \quad y' = y - l \end{align}

We should acknowledge this is relatable in 3-space too such that the translation equations are as follows:

(2)
\begin{align} \quad x' = x - k \quad y' = y - l \quad z' = z - m \end{align}

## Example 1

Suppose we have an $xy$-coordinate system be translated to obtain an $x'y'$-coordinate system that has it's origin at $xy$-coordinates $(2, 3)$. Determine the $x'y'$-coordinates of the point with the $xy$-coordinates $(-2, 1)$.

First let's set up our translation equations:

(3)
\begin{align} x' = x - 2 \quad y' = y - 3 \end{align}

We can now substitute appropriate to solve for $x'$ and $y'$:

(4)
\begin{align} x' = -2 -2 \quad y' = 1 - 3 \end{align}
(5)
\begin{align} x' = -4 \quad, y' = -2 \end{align}

Thus the $x'y'$-coordinates that correspond to the axes translation for the $xy$-coordinates $(-2, 1)$ are $(-4, -2)$.