vectors - Vektoren
All arrows with the same length and direction belong to the same class of vectors (see example of boats in the same wind in the Cornelsen book MA-3).
notation - Schreibweise
We generally use minuscules with an arrow on top.
${\vec{a}}$, ${\vec{b}}$, ${\vec{c}}$, …
A vector can also be described using the endpoints. Here, ${\overrightarrow{P_1 P_2}}$ where the vector is pointing from $P_1$ to $P_2$.
${\overrightarrow{A_1 A_2}}$, ${\overrightarrow{B_1 B_2}}$, ${\overrightarrow{C_1 C_2}}$, …
In international text books, vectors may also be written in bold.
example
${\textbf{i}}$ , ${\textbf{j}}$ and ${\textbf{k}}$ are bold.
In the old days, when handwritten text was given to a printer, to show the typesetter what text was to be bold that particular text was underlined.
So, occasionally you may come across ${\underline{a}}$, ${\underline{b}}$ and ${\underline{c}}$ in handwritten manuscripts.
column vectors - Spaltenvektoren
translation - Verschiebungsvektor
From P to Q the vector ${\vec{v}}$ may be represented as a column vector. Here, we calculate the difference of corresponding coordinates.
example
A vector $\vec{v}$ pointing from P(2,4) to Q(7,1) is
${\vec{v} = \overrightarrow{P Q} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix} = \begin{pmatrix} 7 - 2 \\ 1 - 4 \end{pmatrix} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}}$
The same also works in space if the coordinates of the points are known.
exercise
TODO
position vectors - Ortsvektor
The position vector ${\overrightarrow{O P}}$ is a vector drawn from the origin O to a point $P(p_1,p_2)$ or $P(p_1,p_2,p_3)$ in space. In German - Ortsvektor.
${\vec{p} = \overrightarrow{O P} = \begin{pmatrix} p_1 \\ p_2 \end{pmatrix} }$or
${\vec{p} = \overrightarrow{O P} = \begin{pmatrix} p_1 \\ p_2 \\ p_3 \end{pmatrix} }$, respectively.
example
${\vec{p} = \overrightarrow{O P} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} }$
magnitude - Betrag
The magnitude of a vector ${\vec{a}}$ is written as
${|\vec{a}|}$ and is the length of the vector.
Say,
${\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix} }$. Then,${|\vec{v}| = \sqrt{x^2 + y^2}}$.
Similarly, for
${\vec{v} = \begin{pmatrix} x \\ y \\ z\end{pmatrix} }$the magnitude${|\vec{v}| = \sqrt{x^2 + y^2 + z^2}}$.
example
${ \left| \begin{pmatrix} 4 \\ 3 \end{pmatrix} \right| = \sqrt{4^2 + 3^2}= \sqrt{25} = 5}$
${ \left| \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \right| = \sqrt{2^2 + 3^2 + 4^2}= \sqrt{29} \approx{5.3851}}$