# mutual positions of vectors

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

1. 1.

Two non-zero Euclidean vectors $\vec{a}$ and $\vec{b}$ are said to be , denoted by  $\vec{a}\parallel\vec{b}$,  iff there exists a real number $k$ such that

 $\vec{a}=k\vec{b}.$

Since both $\vec{a}$ and $\vec{b}$ are non-zero, $k\neq 0$.  So $\parallel$ is a binary relation on on  $E\!\smallsetminus\!\{\vec{0}\}$  and called the parallelism.  If  $k>0$,  then $a$ and $b$ are said to be in the same direction, and we denote this by  $\vec{a}\upuparrows\vec{b}$;  if  $k<0$,  then $a$ and $b$ are said to be in the opposite or contrary directions, and we denote this by  $\vec{a}\downarrow\uparrow\vec{b}$.

Remarks

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

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

2. 2.

Two Euclidean vectors $\vec{a}$ and $\vec{b}$ are , denoted by  $\vec{a}\perp\vec{b}$,  iff

 $\vec{a}\cdot\vec{b}=0,$

i.e. iff their scalar product vanishes.  Then $\vec{a}$ and $\vec{b}$ are normal vectors of each other.

Remarks

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

• 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. 3.

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

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

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