Characteristics of Vector in Plane/Examples/-y, x

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Examples of Use of Characteristics of Vector in Plane

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


The ordered pair $\tuple {-y, x}$ can be interpreted as the components of a position vector.


Proof

We use Characteristics of Vector in Plane.

Let:

\(\ds {V'}_x\) \(=\) \(\ds -y \cos \varphi + x \sin \varphi\)
\(\ds {V'}_y\) \(=\) \(\ds y \sin \varphi + x \cos \varphi\)

by setting $V_x = -y$ and $V_y = x$.

From Characteristics of Vector in Plane:

\(\ds {V'}_x\) \(=\) \(\ds -y'\)
\(\ds {V'}_y\) \(=\) \(\ds x'\)

Using Rotation of Cartesian Axes around Vector it is seen that the equalities hold.

Hence the result.


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