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Vector coefficients

We start this section with a definition.

Definition (D 1.4.1) A set of 3 linearly independent vectors r$_1$, r$_2$, and r$_3$ in space is called a basis of the vector space. Any vector r of the vector space can be written in the form . The vectors r$_1$, r$_2$, and r$_3$ are called basis vectors; the vector r is called a linear combination of r$_1$, r$_2$, and r$_3$. The real numbers $\lambda_1$, $\lambda_2$, and $\lambda_3$ are called the coefficients of r with respect to the basis r$_1$, r$_2$, r$_3$. In crystallography the 2 basis vectors for the plane are mostly called a and b or $\mathbf{a}_1$ and $\mathbf{a}_2$, and the 3 basis vectors of the space are a, b, and c or $\mathbf{a}_1$, $\mathbf{a}_2$, and $\mathbf{a}_3$.

The vector $\stackrel{\longrightarrow}{XY}=\mathbf{r}$ connects the points $X$ and $Y$, see Fig. 1.3.1. In Section 1.1 the coordinates $x$, $y$, and $z$ of a point $P$ have been introduced, see Fig. 1.1.1. We now replace the section $(OA)$ on the coordinate axis $a$ by the vector $\stackrel{\longrightarrow}{OA}\ =\mathbf{a}$, $(OB)$ on $b$ by $\stackrel{\longrightarrow}{OB}\ =\mathbf{b}$, and $(OC)$ on $c$ by $\stackrel{\longrightarrow}{OC}\ =\mathbf{c}$. If $X$ and $Y$ are given by their columns of coordinates with respect to these coordinate axes, then the vector $(\stackrel{\longrightarrow} {XY})$ is determined by the column of the three coordinate differences between the points $X$ and $Y$. These differences are the vector coefficients of r:

$$\mbox{\textit{\textbf{r}}}=\left(\begin{array}{c} y_1-x_1 \\ ... ...left( \begin{array}{c} y_1 \\ y_2 \\ y_3 \end{array} \right).$$ (1.4.1)

As the point coordinates, the vector coefficients are written in a column. It is not always obvious whether a column of 3 numbers represents a point by its coordinates or a vector by its coefficients. One often calls this column itself a `vector'. However, this terminology should be avoided. In crystallography both, points and vectors are considered. Therefore, a careful distinction between both items is necessary.

An essential difference between the behaviour of vectors and points is provided by the changes in their coefficients and coordinates if another origin $O'$ in point space is chosen:

Let $O'$ be the new, $O$ the old origin, and o$'$ the column of coordinates of $O'$ with respect to the old coordinate system: .

Then and the coordinates of $X$ and $Y$ in the old coordinate system, are replaced by the columns and of the coordinates in the new coordinate system, see Fig.1.4.1.

From $\mathbf{x}=\mathbf{o}'+\mathbf{x}'$ follows $x_1=o_1' + x_1'$ and $x_1' = x_1-o_1'$, etc. Therefore, the coordinates of the points change if one chooses a new origin.

However, the coefficients of the vector $(\stackrel{\longrightarrow} {XY})$ do not change because of $y_1'-x_1' = y_1-o_1'-(x_1-o_1') = y_1-x_1$, etc.

Fig. 1.4.1 The coordinates of the points $X$ and $Y$ with respect to the old origin $O$ are x and y, with respect to the new origin $O'$ are $\mbox{\textit{\textbf{x}}}'$ and $\mbox{\textit{\textbf{y}}}'$. From the diagram one reads the equations $\mbox{\textit{\textbf{o}}}' + \mbox{\textit{\textbf{x}}}' = \mbox{\textit{\textbf{x}}}$ and $\mbox{\textit{\textbf{o}}}' + \mbox{\textit{\textbf{y}}}' = \mbox{\textit{\textbf{y}}}.$

The rules 1., 2., and 5. of Section 1.3 (the others are then obvious) are expressed by:

1. The vector x is multiplied with a real number $\lambda$
   
   $$\lambda\,\mbox{\textit{\textbf{x}}}=\lambda\left( \begin{arra... ...da\,x_1 \\ \lambda \,x_2 \\ \lambda\,x_3 \end{array} \right).$$
   
   
2. For successive multiplication of x with $\lambda$ and $\mu$
   
   $$\mu(\lambda\,\mbox{\textit{\textbf{x}}})=\mu \left( \begin{ar... ...\ \mu\,\lambda\,x_2 \\ \mu\,\lambda\,x_3 \end{array} \right).$$
   
   
3. The sum $\mathbf{z}=\mathbf{x}+\mathbf{y}$ of two vectors is calculated by their columns x and y
   
   $$\mbox{\textit{\textbf{z}}}=\left(\hspace{-0.4em} \begin{array... ...1+x_1 \\ y_2+x_2 \\ y_3+x_3 \end{array}\hspace{-0.4em} \right).$$
   
   

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