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Mathematics-Online course: Preparatory Course Mathematics - Linear Algebra and Geometry - Vectors

Vectors in the Three-Dimensional Space

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A vector is a directed line segment. The vector represented by the arrow from a point $ P$ to a point $ Q$ is denoted by $ \overrightarrow{PQ} .$

Two vectors are regarded as equal when they have the same length and direction. $ \overrightarrow{P_1Q_1} = \overrightarrow{P_2Q_2} $ means that the directed line segments from $ P_1$ to $ Q_1$ and $ P_2$ to $ Q_2$ represent the same vector.

A vector represents a parallel translation of the space:

$\displaystyle \vec{a} = \overrightarrow{PQ} $

corresponds to the translation of the point $ P$ into the point $ Q$ , pictured by an arrow.
\includegraphics[width=6.4cm]{a_vec_raum}
As the figure shows, arrows with the same direction and length represent the same vector. The special representation with respect to the origin of the coordinate sytem is called position vector and defines the vector's coordinates:

$\displaystyle \overrightarrow{OA} = \left(\begin{array}{c}a_1\\ a_2\\ a_3\end{array}\right)\, . $

The coordinates of $ \vec{a}$ can also be obtained by calculating the difference of the recpective coordinates of the points $ P$ and $ Q$ . Finally, the zero vector is denoted by

$\displaystyle \overrightarrow{OO} = \overrightarrow{PP} = \vec{0}\,. $

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  automatically generated 1/9/2017