mutual positions of vectors
In this entry, we work within a Euclidean space $E$.

1.
Two nonzero Euclidean vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are said to be parallel^{}, denoted by $\overrightarrow{a}\parallel \overrightarrow{b}$, iff there exists a real number $k$ such that
$$\overrightarrow{a}=k\overrightarrow{b}.$$ Since both $\overrightarrow{a}$ and $\overrightarrow{b}$ are nonzero, $k\ne 0$. So $\parallel $ is a binary relation^{} on on $E\setminus \{\overrightarrow{0}\}$ and called the parallelism. If $k>0$, then $a$ and $b$ are said to be in the same direction, and we denote this by $\overrightarrow{a}\upuparrows \overrightarrow{b}$; if $$, then $a$ and $b$ are said to be in the opposite or contrary directions, and we denote this by $\overrightarrow{a}\downarrow \uparrow \overrightarrow{b}$.
Remarks

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Actually, the parallelism is an equivalence relation^{} on $E\setminus \{\overrightarrow{0}\}$. If the zero vector^{} $\overrightarrow{0}$ were allowed along, then the relation^{} were not symmetric^{} ($\overrightarrow{0}=0\overrightarrow{b}$ but not necessarily $\overrightarrow{b}=k\overrightarrow{0}$).

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When two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are not parallel to one another, written $\overrightarrow{a}\nparallel \overrightarrow{b}$, they are said to be diverging.

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2.
Two Euclidean vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular^{}, denoted by $\overrightarrow{a}\u27c2\overrightarrow{b}$, iff
$$\overrightarrow{a}\cdot \overrightarrow{b}=0,$$ i.e. iff their scalar product^{} vanishes. Then $\overrightarrow{a}$ and $\overrightarrow{b}$ are normal vectors of each other.
Remarks

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We may say that $\overrightarrow{0}$ is perpendicular to all vectors, because its direction is and because $\overrightarrow{0}\cdot \overrightarrow{b}=0$.

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Perpendicularity is not an equivalence relation in the set of all vectors of the space in question, since it is neither reflexive^{} nor transitive^{}.

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3.
The angle $\theta $ between two nonzero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is obtained from
$$\mathrm{cos}\theta =\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{\overrightarrow{a}\overrightarrow{b}}.$$ The angle is chosen so that $0\leqq \theta \leqq \pi $.
Title  mutual positions of vectors 
Canonical name  MutualPositionsOfVectors 
Date of creation  20130322 14:36:24 
Last modified on  20130322 14:36:24 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  25 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 15A72 
Related topic  AngleBetweenTwoLines 
Related topic  DirectionCosines 
Related topic  OrthogonalVectors 
Related topic  PerpendicularityInEuclideanPlane 
Related topic  MedianOfTrapezoid 
Related topic  TriangleMidSegmentTheorem 
Related topic  CommonPointOfTriangleMedians 
Related topic  FluxOfVectorField 
Related topic  NormalOfPlane 
Defines  parallel 
Defines  parallelism 
Defines  perpendicular 
Defines  perpendicularity 
Defines  diverging 
Defines  normal vector 