Difference between revisions of "Vector algebra"
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{{Navigation|before=[[Table of contents]]|overview=[[Vector algebra]]|next=[[Unit vectors]]}} | {{Navigation|before=[[Table of contents]]|overview=[[Vector algebra]]|next=[[Unit vectors]]}} | ||
This article gives a brief explanation of vectors and vector algebra. <br/> | This article gives a brief explanation of vectors and vector algebra. <br/> | ||
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A scalar value is just a numeric value. In contrast thereto, a vectorial value has a direction. Examples for vectors are all forces and the velocity, which is directed to the driving direction. Vectors are usually denoted with an arrow, for example <math>\vec{\mathbf{a}}</math>. A three-dimensional vector for example is<br/><br/> | A scalar value is just a numeric value. In contrast thereto, a vectorial value has a direction. Examples for vectors are all forces and the velocity, which is directed to the driving direction. Vectors are usually denoted with an arrow, for example <math>\vec{\mathbf{a}}</math>. A three-dimensional vector for example is<br/><br/> | ||
<math>\vec{\mathbf{a}}= | <math>\vec{\mathbf{a}}= | ||
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where <math>a_x</math>, <math>a_y</math> and <math>a_z</math> describe the components of the vector in x-, y- and z-direction. For graphical representations arrows are used that show the direction of the vector and whose length equals its magnitude. The magnitude of a vector is defined as<br/><br/> | where <math>a_x</math>, <math>a_y</math> and <math>a_z</math> describe the components of the vector in x-, y- and z-direction. For graphical representations arrows are used that show the direction of the vector and whose length equals its magnitude. The magnitude of a vector is defined as<br/><br/> | ||
<math>|\vec{\mathbf{a}}|=\sqrt{a_x^2+a_y^2+a_z^2}</math><br/><br/> | <math>|\vec{\mathbf{a}}|=\sqrt{a_x^2+a_y^2+a_z^2}</math><br/><br/> | ||
| + | [[File:vectorsopposite.png|right|350px]] | ||
As the presented notation is comparatively sophisticated <math>|\vec{\mathbf{a}}|</math> is often reduced to <math>a</math>. The vector <math>-\vec{\mathbf{a}}</math> has the same magnitude as <math>\vec{\mathbf{a}}</math> but is directed opposite. Two or more vectors are called equal if they have the same direction and the same magnitude. The zero vector <math>\vec{\mathbf{0}}</math> is a special vector with undefined direction and magnitude zero. | As the presented notation is comparatively sophisticated <math>|\vec{\mathbf{a}}|</math> is often reduced to <math>a</math>. The vector <math>-\vec{\mathbf{a}}</math> has the same magnitude as <math>\vec{\mathbf{a}}</math> but is directed opposite. Two or more vectors are called equal if they have the same direction and the same magnitude. The zero vector <math>\vec{\mathbf{0}}</math> is a special vector with undefined direction and magnitude zero. | ||
Revision as of 15:22, 14 May 2014
| ← Back: Table of contents | Overview: Vector algebra | Next: Unit vectors → |
This article gives a brief explanation of vectors and vector algebra.
A scalar value is just a numeric value. In contrast thereto, a vectorial value has a direction. Examples for vectors are all forces and the velocity, which is directed to the driving direction. Vectors are usually denoted with an arrow, for example 
. A three-dimensional vector for example is
![\vec{\mathbf{a}}=
\left[\begin{array}{cccc}
a_x\\
a_y\\
a_z
\end{array}\right]](/wiki/robotics/images/math/5/0/f/50f88ee28302078446e3f5fa93e30769.png)
where 
, 
and 
describe the components of the vector in x-, y- and z-direction. For graphical representations arrows are used that show the direction of the vector and whose length equals its magnitude. The magnitude of a vector is defined as
As the presented notation is comparatively sophisticated 
is often reduced to 
. The vector 
has the same magnitude as but is directed opposite. Two or more vectors are called equal if they have the same direction and the same magnitude. The zero vector 
is a special vector with undefined direction and magnitude zero.
