# Vector Components

## Circle with Points

Details

The given components $$(v_x, v_y)$$ define 5 points on a circle with radius $$r=\sqrt{v_{x}^2 + v_{y}^2}$$:

 Point Position $$\left( v_{\bar x} , v_{\bar y} \right)$$ 1 given $$(v_x, v_y)$$ 2 3 o’clock $$(r, 0)$$ 3 9 o’clock $$(-r, 0)$$ 4 12 o’clock $$(0, r)$$ 5 6 o’clock $$(0, -r)$$

Rules

• Find point 1 at the end of the red line. Read the vector components $$(v_x, v_y)$$ as the point coordinates of point 1.

• Drag the grey boxes to where they belong.

• Check your result by checking the 5 points.

Example = First image: $$(v_x, v_y) = (3, 4)$$.

The circle radius is $$r=\sqrt{v_{x}^2 + v_{y}^2}=5$$.

 Point Position $$\left( v_{\bar x} , v_{\bar y} \right)$$ 1 given $$(3, 4)$$ 2 3 o’clock $$(5, 0)$$ 3 9 o’clock $$(-5, 0)$$ 4 12 o’clock $$(0, 5)$$ 5 6 o’clock $$(0, -5)$$

## $$\varphi = \varphi_1 \Leftrightarrow v_{\bar x}$$ maximal

Details

Find the angle $$\varphi=\varphi_1$$, for which $$v_{\bar x}$$ is maximal.

 Angle counted from to $$\varphi_1$$ $$\circlearrowright$$ $$\color{red}{｜}$$ $$\color{red}{￤}$$

Rules

• Read $$(v_x, v_y)$$ as the point coordinates of point 1 at the end of the solid red line.

• Find $$r=\sqrt{v_{x}^2 + v_{y}^2}$$.

• Find $$\varphi_1$$ using 1 :

$\begin{split}\tan \frac{\varphi_1}{2}&=\frac{v_y}{v_x + r }\\ \varphi_1 &=2 \arctan \frac{v_y}{v_x + r }\end{split}$

And find this angle in the diagram.

• Drag the grey boxes to where they belong.

Example: $$(v_x, v_y) = (3, 4)$$ leads to $$r=5$$ and $$\varphi_1\approx 53^\circ$$.

## Passive Transformation

Details

Use:

$\begin{split}\begin{bmatrix} v_{\bar x} \\ v_{\bar y} \end{bmatrix}= \begin{bmatrix} c_\varphi & s_\varphi \\ -s_\varphi & c_\varphi \end{bmatrix} \begin{bmatrix} v_x \\ v_y \end{bmatrix}\end{split}$
 Winkel Zählrichtung von bis zu $$\varphi$$ $$\circlearrowright$$ $$\color{red}{｜}$$ $$\color{red}{￤}$$ $$\varphi$$ $$\circlearrowleft$$ $$\color{blue}{￤}$$ $$\color{blue}{｜}$$

Rules

• Read $$(v_x, v_y)$$ as the point coordinates of point 1 at the end of the solid red line.

• Read $$\varphi$$.

• Calculated $$(v_{\bar x},v_{\bar y})$$ for this angle.

• Drag the grey boxes to where they belong.

Example: First image:

• $$(v_x, v_y) = (3, 4)$$

• $$\varphi=30^\circ$$.

• $\begin{split}\begin{bmatrix} v_{\bar x} \\ v_{\bar y} \end{bmatrix} &= \begin{bmatrix} \cos 30^\circ & \sin 30^\circ \\ -\sin 30^\circ & \cos 30^\circ \end{bmatrix} \begin{bmatrix} v_x \\ v_y \end{bmatrix} \\ &\approx \begin{bmatrix} 4.60 \\ 1.96 \end{bmatrix}\end{split}$

## Active Transformation

Details

Use:

$\begin{split}\begin{bmatrix} v'_{x} \\ v'_{y} \end{bmatrix} = \begin{bmatrix} c_\alpha & -s_\alpha \\ s_\alpha & c_\alpha \end{bmatrix} \begin{bmatrix} v_x \\ v_y \end{bmatrix}\end{split}$
 Angle Counting from to $$-\alpha$$ $$\circlearrowright$$ $$\color{red}{｜}$$ $$\color{red}{￤}$$ $$\alpha$$ $$\circlearrowleft$$ $$\color{red}{｜}$$ $$\color{red}{￤}$$

Rules

• Read $$(v_x, v_y)$$ as the point coordinates of point 1 at the end of the solid red line.

• Read $$\alpha$$.

• Calculate $$(v'_x, v'_y)$$.

• Drag the grey boxes to where they belong.

Example: First image:

• $$(v_x, v_y) = (3, 4)$$

• $$-\alpha=30^\circ$$ so that $$\alpha=-30^\circ$$

• $\begin{split}\begin{bmatrix} v'_x \\ v'_y \end{bmatrix} &= \begin{bmatrix} \cos( -30^\circ) & -\sin (-30^\circ) \\ \sin( -30^\circ) & \cos (-30^\circ) \end{bmatrix} \begin{bmatrix} v_x \\ v_y \end{bmatrix} \\ &= \begin{bmatrix} \cos 30^\circ & \sin 30^\circ \\ -\sin 30^\circ & \cos 30^\circ \end{bmatrix} \begin{bmatrix} v_x \\ v_y \end{bmatrix} \\ &\approx \begin{bmatrix} 4.60 \\ 1.96 \end{bmatrix}\end{split}$

## Test

Footnotes:

1

Note the periodicity: The domain of the angles in the grey boxes is $$-180^\circ < \varphi_1 \le 180^\circ$$. Also your calculated will most likely return angles in this interval. However: The angles in the diagrams are angles in the domain $$0^\circ \le \varphi_1 \le 360^\circ$$ so that all angles are positive.