mutual positions of vectors

In this entry, we work within a Euclidean space $E$.

1. Two non-zero 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 non-zero, $k\ne 0$.  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

   • 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  $\overrightarrow{b}=k\overrightarrow{0}$).

   • 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.

2. Two Euclidean vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular, denoted by  $\overrightarrow{a}⟂\overrightarrow{b}$,  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

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

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

3. The angle $\theta$ between two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is obtained from

   $$\cos \theta =\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}.$$

   The angle is chosen so that  $0\leqq \theta \leqq \pi$.

Title	mutual positions of vectors
Canonical name	MutualPositionsOfVectors
Date of creation	2013-03-22 14:36:24
Last modified on	2013-03-22 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