Characteristics of Vector in Plane

Theorem
Let a Cartesian plane $\CC$ be established with origin $O$.

Let $\tuple {A_x, A_y}$ be an ordered pair of real numbers that can be used to represent a point in $\CC$.

Then:
 * $\tuple {A_x, A_y}$ are the Cartesian coordinates of the terminal point of a position vector $\mathbf A$


 * $\tuple {A_x, A_y}$ can be transformed into $\tuple { {A'}_x, {A'}_y}$ by rotating $\CC$ about $O$ through an angle $\varphi$ using:
 * $\tuple {A_x, A_y}$ can be transformed into $\tuple { {A'}_x, {A'}_y}$ by rotating $\CC$ about $O$ through an angle $\varphi$ using:

Sufficient Condition
Let $\tuple {A_x, A_y}$ represent the terminal point of a position vector $\mathbf A$.

Then from Rotation of Cartesian Axes around Vector:

Necessary Condition
Let $\tuple {A_x, A_y}$ fulfil the condition that it can be transformed into $\tuple { {A'}_x, {A'}_y}$ by rotating $\CC$ about $O$ through an angle $\varphi$ to $\CC'$ using the given equations.

Let the components $\tuple {A_x, A_y}$ of $\mathbf A$ be functions of the coordinates and perhaps of some other constant vector $\mathbf c$:

In the rotated plane $\CC'$, $\mathbf A$ has components $\tuple { {A'}_x, {A'}_y}$ which are also functions of the same things:

From Rotation of Cartesian Axes around Vector, the values $\tuple {x', y', {c'}_x, {c'}_y}$ can be replaced by $\tuple {x, y, c_x, c_y}$ and the angle of rotation $\varphi$.

In the special case where $\varphi = 0$, we have:

and so on.

It follows that:

Hence ${A'}_x$ is the same function of $\tuple {x', y', {c'}_x, {c'}_y}$ as $A_x$ is of $\tuple {x, y, c_x, c_y}$.

Similarly for ${A'}_y$ and $A_y$.