# Vector Components

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## Used Symbols

 counts pos. about shows angular pos. of relative to $$\varphi$$ $$z=\bar z$$ $$(\bar x, \bar y)$$ $$(x, y)$$ $$\alpha$$ $$z=\bar z$$ $$\boldsymbol v'$$ $$\boldsymbol v$$
 $$(x,y)$$-Comp. $$(\bar x, \bar y)$$-Comp. $$\boldsymbol v$$ $$\begin{bmatrix} v_{x}\\ v_{y} \end{bmatrix}$$ $$\begin{bmatrix} v_{\bar x} \\ v_{\bar y} \end{bmatrix}$$ $$\boldsymbol v'$$ $$\begin{bmatrix} v'_{x} \\ v'_{y} \end{bmatrix}$$ undefined

## Passive und aktive Transformation Passive transformation (on the left) und active transformation (on the right) for the special case that $$-\alpha=\varphi$$, so that $$(v_{\bar x}, v_{\bar y}) = (v'_{x}, v'_{y})$$.

Passive und aktive Transformation

For a vector given by its $$(x,y)$$-components wrt an $$(x,y)$$-frame the following transformations exist:

• Passive transformation: 1 Vector & 2 frames: The $$(\bar x, \bar y)$$-components of the same vector wrt a second frame is calculated. This second frame is the $$(\bar x, \bar y)$$-frame, which is rotated relative to the given $$(x, y)$$-frame.

$\begin{split}\begin{bmatrix} v_{\bar x} \\ v_{\bar y} \end{bmatrix}= \begin{bmatrix} c_\varphi & s_\varphi \\ -s_\varphi & c_\varphi \end{bmatrix} \begin{bmatrix} v_x \\ v_y \end{bmatrix}\end{split}$
• Active transformation: 2 Vectors & 1 frame: The $$(x, y)$$-components of a second vector are calculated. This second vector is rotated relative to the given vector.

$\begin{split}\begin{bmatrix} v'_{x} \\ v'_{y} \end{bmatrix} = \begin{bmatrix} c_\alpha & -s_\alpha \\ s_\alpha & c_\alpha \end{bmatrix} \begin{bmatrix} v_x \\ v_y \end{bmatrix}\end{split}$

## Web-App 1

Vector Transformation

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Vector Components

Footnotes:

1

Unitless Quantities are used. These are defined as the actual quantity divided by its unit: E.g. if a quantity was given in the unit “meter per second”: The respective unitless quantity would be equal to the given quantity divided by “meter per second”.