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:
\(\ds {A'}_x\) | \(=\) | \(\ds A_x \cos \varphi + A_y \sin \varphi\) | ||||||||||||
\(\ds {A'}_y\) | \(=\) | \(\ds -A_x \sin \varphi + A_y \cos \varphi\) |
Proof
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:
\(\ds {A'}_x\) | \(=\) | \(\ds A_x \cos \varphi + A_y \sin \varphi\) | ||||||||||||
\(\ds {A'}_y\) | \(=\) | \(\ds -A_x \sin \varphi + A_y \cos \varphi\) |
$\Box$
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 $\mathbf c$:
\(\ds A_x\) | \(=\) | \(\ds \map {A_x} {x, y, c_x, c_y}\) | ||||||||||||
\(\ds A_y\) | \(=\) | \(\ds \map {A_y} {x, y, c_x, c_y}\) |
In the rotated plane $\CC'$, $\mathbf A$ has components $\tuple { {A'}_x, {A'}_y}$ which are also functions of the same things:
\(\ds {A'}_x\) | \(=\) | \(\ds \map { {A'}_x} {x', y', {c'}_x, {c'}_y}\) | ||||||||||||
\(\ds {A'}_y\) | \(=\) | \(\ds \map { {A'}_y} {x', y', {c'}_x, {c'}_y}\) |
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:
\(\ds A_x\) | \(=\) | \(\ds {A'}_x\) | ||||||||||||
\(\ds A_y\) | \(=\) | \(\ds {A'}_y\) |
and so on.
It follows that:
\(\ds x'\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds y'\) | \(=\) | \(\ds y\) |
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 $A_y$.
This page needs the help of a knowledgeable authority. In particular: The above has been transliterated directly from Arfken. I don't have a clue what he's talking about. If you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Help}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Examples
Example: $\tuple {-y, x}$
The ordered pair $\tuple {-y, x}$ can be interpreted as the components of a position vector.
Example: $\tuple {x, -y}$
The ordered pair $\tuple {x, -y}$ cannot be interpreted as the components of a position vector.
Sources
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.2$ Rotation of Coordinates