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[parent] mutual positions of vectors (Definition)

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

  1. Two non-zero 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
    $\displaystyle \vec{a} = k\vec{b}.$

    Since both $ \vec{a}$ and $ \vec{b}$ are non-zero, $ k\neq 0$. So $ \parallel$ is a binary relation on on $ E\!\smallsetminus\!\lbrace\vec{0}\rbrace$ 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

  2. Two Euclidean vectors $ \vec{a}$ and $ \vec{b}$ are perpendicular, denoted by $ \vec{a}\perp\vec{b}$, iff
    $\displaystyle \vec{a}\cdot\vec{b} = 0,$
    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 non-zero vectors $ \vec{a}$ and $ \vec{b}$ is obtained from
    $\displaystyle \cos\theta = \frac{\vec{a}\cdot\vec{b}}{\vert\vec{a}\vert\vert\vec{b}\vert}.$
    The angle is chosen so that $ 0 \leqq \theta \leqq \pi$.



"mutual positions of vectors" is owned by pahio. [ full author list (2) ]
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See Also: angle between two lines, direction cosines, orthogonal vectors, perpendicularity in Euclidean plane, median of trapezoid, triangle mid-segment theorem, common point of triangle medians

Also defines:  parallel, parallelism, perpendicular, perpendicularity, diverging, normal vector
Keywords:  angle between two vectors

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right-handed system of vectors (Definition) by juanman
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Cross-references: non-zero vectors, angle, transitive, Reflexive, vanishes, scalar product, vectors, symmetric, relation, zero vector, equivalence relation, opposite, binary relation, real number, iff, Euclidean vectors, Euclidean space
There are 67 references to this entry.

This is version 22 of mutual positions of vectors, born on 2004-09-16, modified 2007-08-28.
Object id is 6178, canonical name is MutualPositionsOfVectors.
Accessed 12213 times total.

Classification:
AMS MSC: 15A72 (Linear and multilinear algebra; matrix theory :: Vector and tensor algebra, theory of invariants)
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