Difference between revisions of "Unit vectors"
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{{Navigation|before=[[Vector algebra]]|overview=[[Vector algebra]]|next=[[Simple arithmetic operations]]}} | {{Navigation|before=[[Vector algebra]]|overview=[[Vector algebra]]|next=[[Simple arithmetic operations]]}} | ||
+ | <table style="width:100%"><td style="width:50%">{{Exercise|Selftest: Unit vector}}</td><td style="width:50%">{{Matlab|MATLAB: Unit vectors}}</td></table> | ||
− | A unit vector is a vector with magnitude 1. The unit vector to a given vector <math>\vec{\mathbf{a}}</math> can be | + | A unit vector is a vector with magnitude 1. The unit vector to a given vector <math>\vec{\mathbf{a}}</math> can be determined by dividing the given vector by its magnitude <math>|\vec{\mathbf{a}}|</math>: |
+ | [[File:unitvector.png|right|300px]] | ||
:<math> | :<math> | ||
\vec{\textbf{e}}_{a} = \frac{\vec{\textbf{a}}}{|\vec{\textbf{a}}|} = | \vec{\textbf{e}}_{a} = \frac{\vec{\textbf{a}}}{|\vec{\textbf{a}}|} = | ||
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:<math> | :<math> | ||
\vec{\textbf{a}} = \frac{\vec{\textbf{a}}}{|\vec{\textbf{a}}|} |\vec{\textbf{a}}| = \vec{\textbf{e}}_{a} |\vec{\textbf{a}}| | \vec{\textbf{a}} = \frac{\vec{\textbf{a}}}{|\vec{\textbf{a}}|} |\vec{\textbf{a}}| = \vec{\textbf{e}}_{a} |\vec{\textbf{a}}| | ||
+ | </math><br/> | ||
+ | How arithmetic operations like fractions are handled exactly, is described in the article about [[Simple arithmetic operations|simple arithmetic operations]]. | ||
+ | |||
+ | {{ExampleMatlab | ||
+ | |Title=Determination of the unit vector to a given vector | ||
+ | |Matlab=[[MATLAB: Unit vectors|see MATLAB code]] | ||
+ | |Contents= | ||
+ | |||
+ | ''Hint: For detailed information about the handling of arithmetic operations please have a look on the article about [[Simple arithmetic operations|simple arithmetic operations]]'' | ||
+ | |||
+ | To the given vector <math>\vec{\mathbf{b}}</math> the corresponding unit vector shall be determined:<br/> | ||
+ | :<math> | ||
+ | \vec{\mathbf{b}}=\begin{bmatrix} 3\\ 0\\ 4 \end{bmatrix} | ||
+ | </math> | ||
+ | :<math> | ||
+ | \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} | ||
</math> | </math> | ||
+ | The calculation of the magnitude shows thats it equals 1 indeed: | ||
+ | :<math> | ||
+ | |\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 | ||
+ | </math> | ||
+ | }} | ||
+ | |||
+ | <noinclude>==Literature== | ||
+ | * Manfred Albach, ''Grundlagen der Elektrotechnik 1: Erfahrungssätze, Bauelemente, Gleichstromschaltungen'', 3. Edition (Pearson Studium, 2011) | ||
+ | </noinclude> | ||
+ | |||
+ | [[Category:Article]] | ||
+ | [[Category:Vectors]] |
Latest revision as of 18:02, 13 November 2015
← Back: Vector algebra | Overview: Vector algebra | Next: Simple arithmetic operations → |
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A unit vector is a vector with magnitude 1. The unit vector to a given vector can be determined by dividing the given vector by its magnitude :
The vector has the magnitude 1 (so ) and is pointed to the direction of . So every vector can be described by its magnitude (so a scalar value) and the corresponding unit vector. Therefore can also be written as follows:
How arithmetic operations like fractions are handled exactly, is described in the article about simple arithmetic operations.
Hint: For detailed information about the handling of arithmetic operations please have a look on the article about simple arithmetic operations To the given vector the corresponding unit vector shall be determined: The calculation of the magnitude shows thats it equals 1 indeed: |
Literature
- Manfred Albach, Grundlagen der Elektrotechnik 1: Erfahrungssätze, Bauelemente, Gleichstromschaltungen, 3. Edition (Pearson Studium, 2011)