You are here
Homemutual positions of vectors
Primary tabs
mutual positions of vectors
In this entry, we work within a Euclidean space $E$.
1. Two nonzero Euclidean vectors $\vec{a}$ and $\vec{b}$ are said to be parallel, 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 nonzero, $k\neq 0$. 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 $\vec{b}=k\vec{0}$).

2. Two Euclidean vectors $\vec{a}$ and $\vec{b}$ are perpendicular, 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 indefinite 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. The angle $\theta$ between two nonzero 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 $0\leqq\theta\leqq\pi$.
Mathematics Subject Classification
15A72 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections