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]]}} | ||
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| + | A unit vector is a vector with magnitude 1. The unit vector to a given vector <math>\vec{\mathbf{a}}</math> can be determinde by dividing the given vector by its magnitude <math>|\vec{\mathbf{a}}|</math>: | ||
:<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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\frac{1}{\sqrt{a_x^2 + a_y^2 + a_z^2}} \begin{bmatrix} a_x\\ a_y\\ a_z \end{bmatrix} | \frac{1}{\sqrt{a_x^2 + a_y^2 + a_z^2}} \begin{bmatrix} a_x\\ a_y\\ a_z \end{bmatrix} | ||
</math> | </math> | ||
| − | + | The vector <math>\vec{\mathbf{e}}_{a}</math> has the magnitude 1 (so <math>|\vec{\mathbf{e}}_a|=1</math>) and is pointed to the direction of <math>\vec{\mathbf{a}}</math>. So every vector can be described by its magnitude (so a scalar value) and the corresponding unit vector. Therefore <math>\vec{\mathbf{a}}</math> can also be written as follows: | |
:<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> | </math> | ||
Revision as of 16:02, 14 May 2014
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A unit vector is a vector with magnitude 1. The unit vector to a given vector 
can be determinde 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:
