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Vectors

Vectors are objects which are encountered everywhere in crystallography: as distance vectors between atoms, as basis vectors of the coordinate system, as translation vectors of a crystal lattice, as vectors of the reciprocal lattice, etc. They are elements of the vector space which is studied by linear algebra and is an abstract space. However, vectors can be interpreted easily visually, see Fig.1.3.1:





\psfig{figure=vec1.eps,width=4cm}
Fig. 1.3.1 Vector $(\stackrel{\longrightarrow}
{XY})$ from point X to point Y. The vector represented by an arrow depends only on the relative but not on the absolute sites of the points. The 4 parallel arrows represent the same vector.


For each pair of points X and Y one can draw the arrow $\stackrel{\longrightarrow}{XY}$ from X to Y. The arrow $\stackrel{\longrightarrow}{XY}$ is a representation of the vector r, as is any arrow of the direction and length of r, see Fig. 1.3.1. The set of all vectors forms the vector space. The vector space has no origin but instead there is the zero vector or o vector $(\stackrel{\longrightarrow}{XX})$ which is obtained by connecting any point $X$ with itself. The vector r has a length which is designed by $\vert\mathbf {r}\vert = r$, where r is a non-negative real number. This number is also called the absolute value of the vector. A formula for the calculation of $r$ can be found in Sections 1.6 and 2.6.

For such vectors some simple rules hold which can be visualized, e.g. by a drawing in the plane:

  1. If $\lambda$ is a real number then the vector $\lambda\,\mathbf{r}=\mathbf{r}\,\lambda$ is defined as the vector parallel to r and with length $\vert\lambda\,\mathbf{r}\vert= \lambda\,\vert\mathbf{r}\vert=\lambda\,r$.

    In particular, $(1/\textit{r})\,\mathbf{r}=\mathbf{r}_{\circ}$ is a vector of length 1. Such a vector is called a unit vector. Further $1\,\mathbf{r}=\mathbf{r}$; $0\,\mathbf{r}=
\mathbf{o}$ is the zero-vector with length 0. It is the only vector with no direction. $(-1)\,\mathbf{r}=-\mathbf{r}$ is that vector which has the same length as r, $\vert\mathbf{r}\vert = \vert\mathbf{-r}\vert$, but opposite direction.

  2. For successive multiplication with the real numbers $\lambda$ and $\mu$, the relation
    $\mu\,(\lambda\,\mathbf{r})=(\mu\,\lambda)\,\mathbf{r}$ holds.
  3. For 2 real numbers, $\lambda$ and $\mu$, $(\lambda+\mu)\,\mathbf{r}
=\lambda\,\mathbf{r}+\mu\,\mathbf{r}$ holds.
  4. For 2 vectors, r and s, $\lambda\,(\mathbf{r}+\mathbf{s})
=\lambda\,\mathbf{r}+\lambda\,\mathbf{s}$ holds.
  5. For 2 vectors, r and s, $\mathbf{r}+\mathbf{s}=\mathbf{s}+
\mathbf{r}$ holds. This is called the commutative law of vector addition, see Fig.1.3.2 which is also called the parallelogram of forces. In particular, $\mathbf{r}+
(\mathbf{-r})=\mathbf{r}-\mathbf{r}=\mathbf{o}$.
  6. For any 3 vectors, r, s, and t, the associative law of vector addition, see Fig.1.3.3,

    \begin{displaymath}(\textbf{r}+\textbf{s})+\textbf{t}=\textbf{r}+(\textbf{s}+\textbf{t})=
\textbf{r}+\textbf{s}+\textbf{t}\ \
\mbox{holds.}\end{displaymath}



\psfig{figure=vec2.eps,width=5cm}

Fig. 1.3.2 Visualization of the commutative law of vector addition: $\textbf{r} + \textbf{s} = \textbf{s} + \textbf{r}.$

\psfig{figure=vec3.eps,width=5cm} Fig. 1.3.3 Visualization of the associativity of vector addition:

$(\textbf{r} + \textbf{s}) + \textbf{t} = \textbf{r} +
(\textbf{s} + \textbf{t}).$

Definition (D 1.3.2) A set of $n$ vectors $\mathbf{r}_1$, $\mathbf{r}_2$, ..., $\mathbf{r}_n$ is called linearly independent if the equation

\begin{displaymath}\lambda_1\,\mathbf{r}_1+\lambda_2\,\mathbf{r}_2+\ldots
+\lambda_n\,\mathbf{r}_n=0
\end{displaymath} (1.3.1)

can only be fulfilled if $\lambda_1=\lambda_2=\ldots=\lambda_n=0$. Otherwise, the vectors are called linearly dependent.

In the plane any 3 vectors r$_1$, r$_2$, and r$_3$ are linearly dependent because coefficients $\lambda_i$ can always be found such that $\lambda_i$ not all zero and

\begin{displaymath}\lambda_1\,\mathbf{r}_1+\lambda_2\,\mathbf{r}_2+\lambda_3\,\mathbf{r}_3=0
\hspace{3em} \mbox{holds.}\end{displaymath}

Definition (D 1.3.2) The maximal number of linearly independent vectors in a vector space is called the dimension of the space.

As is well known, the dimension of the plane is 2, of the space is 3. Any 4 vectors in space are linearly dependent. Thus, if there are 3 linearly independent vectors r$_1$, r$_2$, and r$_3$, then any other vector r can be represented in the form \( \textbf{r}=\lambda_1\,\textbf{r}_1+\lambda_2\,\textbf{r}_2+\lambda_3\,
\textbf{r}_3.\)

Such a representation is widely used, it will be considered in the next section.


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Next: Vector coefficients Up: Points and vectors Previous: Special coordinate systems: Cartesian coordinates

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