Characteristics of Vector in Plane

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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 $\mathbf A$

if and only if:

$\tuple {A_x, A_y}$ 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 $A_y$.


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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

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