Difference between revisions of "Unit vectors"

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{{Example
 
{{Example
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|Title=Bestimmung des Einheitsvektors zu einem gegebenen Vektor
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|Title=Determination of the unit vector to a given vector
 
|Contents=
 
|Contents=
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Gegeben ist der Vektor <math>\vec{\textbf{b}} = \begin{bmatrix} 3 & 0 & 4 \end{bmatrix}^\text{T}</math> (das <math>^\text{T}</math> steht für [[Transposition]] und ermöglicht die Schreibweise des Spaltenvektors als Zeilenvektor), zu dem der zugehörige Einheitsvektor bestimmt werden soll. In diesem Fall folgt:
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The vector <math>\vec{\mathbf{b}}</math> is given and the corresponding unit vector shall be determined:<br/><br/>
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:<math>
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\vec{\mathbf{b}}=\begin{bmatrix} 3\\ 0\\ 4 \end{bmatrix}
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</math>
 
:<math>
 
:<math>
 
\vec{\textbf{e}}_{b} = \frac{\vec{\textbf{b}}}{|\vec{\textbf{b}}|} =
 
\vec{\textbf{e}}_{b} = \frac{\vec{\textbf{b}}}{|\vec{\textbf{b}}|} =
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\begin{bmatrix} \frac{3}{5}\\ 0\\ \frac{4}{5} \end{bmatrix}
 
\begin{bmatrix} \frac{3}{5}\\ 0\\ \frac{4}{5} \end{bmatrix}
 
</math>
 
</math>
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Das dieser Vektor tatsächlich die Länge 1 hat, lässt sich leicht durch die Bestimmung des Betrags überprüfen:
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The calculation of the magnitude shows thats it equals 1 indeed:
 
:<math>
 
:<math>
 
|\vec{\textbf{e}}_{b}| =
 
|\vec{\textbf{e}}_{b}| =

Revision as of 16:11, 14 May 2014

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A unit vector is a vector with magnitude 1. The unit vector to a given vector \vec{\mathbf{a}} can be determinde by dividing the given vector by its magnitude |\vec{\mathbf{a}}|:

Unitvector.png
\vec{\textbf{e}}_{a} = \frac{\vec{\textbf{a}}}{|\vec{\textbf{a}}|} = \frac{\vec{\textbf{a}}}{\sqrt{a_x^2 + a_y^2 + a_z^2}} = \frac{1}{\sqrt{a_x^2 + a_y^2 + a_z^2}} \begin{bmatrix} a_x\\ a_y\\ a_z \end{bmatrix}

The vector \vec{\mathbf{e}}_{a} has the magnitude 1 (so |\vec{\mathbf{e}}_a|=1) and is pointed to the direction of \vec{\mathbf{a}}. So every vector can be described by its magnitude (so a scalar value) and the corresponding unit vector. Therefore \vec{\mathbf{a}} can also be written as follows:

\vec{\textbf{a}} = \frac{\vec{\textbf{a}}}{|\vec{\textbf{a}}|} |\vec{\textbf{a}}| = \vec{\textbf{e}}_{a} |\vec{\textbf{a}}|
Example: Determination of the unit vector to a given vector

The vector \vec{\mathbf{b}} is given and the corresponding unit vector shall be determined:

\vec{\mathbf{b}}=\begin{bmatrix} 3\\ 0\\ 4 \end{bmatrix}
\vec{\textbf{e}}_{b} = \frac{\vec{\textbf{b}}}{|\vec{\textbf{b}}|} = \frac{1}{\sqrt{3^2 + 0^2 + 4^2}} \begin{bmatrix} 3\\ 0\\ 4 \end{bmatrix} = \frac{1}{\sqrt{25}} \begin{bmatrix} 3\\ 0\\ 4 \end{bmatrix} = \frac{1}{5} \begin{bmatrix} 3\\ 0\\ 4 \end{bmatrix} = \begin{bmatrix} \frac{3}{5}\\ 0\\ \frac{4}{5} \end{bmatrix}

The calculation of the magnitude shows thats it equals 1 indeed:

|\vec{\textbf{e}}_{b}| = \sqrt{\left(\frac{3}{5}\right)^2 + 0^2 + \left(\frac{4}{5}\right)^2} = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{\frac{25}{25}} = \sqrt{1} = 1
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