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.