Vector Components


Used Symbols


counts pos. about

shows angular pos. of

relative to


\(z=\bar z\)

\((\bar x, \bar y)\)

\((x, y)\)


\(z=\bar z\)

\(\boldsymbol v'\)

\(\boldsymbol v\)

Vector Components


\((\bar x, \bar y)\)-Comp.

\(\boldsymbol v\)

\(\begin{bmatrix} v_{x}\\ v_{y} \end{bmatrix}\)

\(\begin{bmatrix} v_{\bar x} \\ v_{\bar y} \end{bmatrix}\)

\(\boldsymbol v'\)

\(\begin{bmatrix} v'_{x} \\ v'_{y} \end{bmatrix}\)


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.

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

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Vector Transformation

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Vector Components



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”.