## 2 Modeling Technique
Angles \(\alpha_{1}\) and \(\alpha_{2}\) are calculated using scalar multiplications:  
\[r_{i\,j}=\sqrt{\left[x-x_{i1}-(j-0.5)\Delta x_{i}\right]^{2}+\left[y-y_{i1}-(j -0.5)\Delta y_{i}\right]^{2}+\left[z-z_{i1}-(j-0.5)\Delta z_{i}\right]^{2}};\]3  
\[r_{i\,j}=\sqrt{\left[x-x_{i1}-(j-0.5)\Delta x_{i}\right]^{2}+\left[y-y_{i1}-(j -0.5)\Delta y_{i}\right]^{2}+\left[z-z_{i1}-(j-0.5)\Delta z_{i}\right]^{2}};\]4  
\[r_{i\,j}=\sqrt{\left[x-x_{i1}-(j-0.5)\Delta x_{i}\right]^{2}+\left[y-y_{i1}-(j -0.5)\Delta y_{i}\right]^{2}+\left[z-z_{i1}-(j-0.5)\Delta z_{i}\right]^{2}};\]5  
The coordinates of the beginning \((x_{a},\;y_{a},\;z_{a})\) of the perpendicular \(r_{i}\) are determined as follows  
\[\left\{ \begin{array}{l}y_{21}(x_{a}-x_{1})=x_{21}(y_{a}-y_{1});\\ z_{21}(y_{a}-y_{1})=y_{21}(z_{a}-z_{1});\\ x_{21}(x_{a}-x)+y_{21}(y_{a}-y)+z_{21}(z_{a}-z)=0,\end{array} \right.\]  
where \(x_{21}=x_{2}-x_{1};y_{21}=y_{2}-y_{1};z_{21}=z_{2}-z_{1}\). The coordinates of the division point of the SW by the perpendicular drawn from \(M\), at \(x_{21}\
eq 0\), are equal to  
\[y_{a}=y_{1}+\frac{y_{21}}{x_{21}}(x_{a}-x_{1});\;z_{a}=z_{1}+\frac{z_{21}}{y_{2 1}}(y_{a}-y_{1})=z_{1}+\frac{z_{21}}{x_{21}}(x_{a}-x_{1});\]  
\[x_{21}(x_{a}-x)+y_{21}\bigg{[}y_{1}+\frac{y_{21}}{x_{21}}(x_{a}-x_{1})-y\bigg{]} +z_{21}\bigg{[}z_{1}+\frac{z_{21}}{x_{21}}(x_{a}-x_{1})-z\bigg{]}=0;\]  
\[x_{21}x_{a}-x_{21}x+y_{21}y_{1}+\frac{y_{21}}{x_{21}}(x_{a}-x_{1})-y_{21}y+z_{ 21}z_{1}+\frac{z_{21}}{x_{21}}(x_{a}-x_{1})-z_{21}z=0;\]  
\[x_{a}=\frac{x_{21}x-y_{21}(y_{1}-y)-z_{21}(z_{1}-z)+x_{1}(y_{21}{}^{2}+z_{21}{ }^{2})/x_{21}}{x_{21}+(y_{21}{}^{2}+z_{21}{}^{2})/x_{21}}.\]  
The following options are possible:  
1. \(x_{21}=0;y_{21}\
eq 0;z_{21}\
eq 0\). \[x_{a}=x_{1};\;y_{a}=\frac{y_{21}z_{21}(z-z_{1})+y_{21}{}^{2}y+z_{21}{}^{2}y_{1 }}{y_{21}{}^{2}+z_{21}{}^{2}};z_{a}=z-\frac{y_{21}}{z_{21}}(y_{a}-y).\]  
2. \(x_{21}=0;y_{21}\
eq 0;z_{21}=0\). \[x_{a}=x_{1};\;y_{a}=y;z_{a}=z_{1}.\]  
3. \(x_{21}=0;y_{21}=0;z_{21}\
eq 0\). \[x_{a}=x_{1};\;y_{a}=y_{1};\;z_{a}=z.\]  
Figure 6: Magnetic field of a short wire. The direction \(\stackrel{{\rightarrow}}{{H}}_{i}\) is determined by vector multiplication, where  
\(\stackrel{{\rightarrow}}{{r}}_{12}=(x_{2}-x_{1})\stackrel{{ \rightarrow}}{{i}}+(y_{2}-y_{1})\stackrel{{\rightarrow}}{{j}}+(z_{ 2}-z_{1})\stackrel{{\rightarrow}}{{k}}\) is the wire axis vector. This can be written as  
\[c_{x}=\frac{(\stackrel{{\rightarrow}}{{r}}_{12}\times\stackrel{{ \rightarrow}}{{r}}_{i1})_{x}}{\left|\stackrel{{ \rightarrow}}{{r}}_{12}\times\stackrel{{\rightarrow}}{{r}}_{i1} \right|}=\frac{c_{m\,x}}{c_{m}};\;c_{y}=\frac{(\stackrel{{ \rightarrow}}{{r}}_{12}\times\stackrel{{\rightarrow}}{{r}}_{i1} )_{y}}{\left|\stackrel{{\rightarrow}}{{r}}_{12}\times\stackrel{{ \rightarrow}}{{r}}_{i1}\right|}=\frac{c_{mz}}{c_{m}};\]  
\[c_{m}=\sqrt{c_{mx}{}^{2}+c_{my}{}^{2}+c_{mz}{}^{2}};c_{mx}=(y_{2}-y_{1})\;(z-z _{1})-(z_{2}-z_{1})\;(y-y_{1});\]  
\[x_{21}(x_{a}-x)+y_{21}\bigg{[}y_{1}+\frac{y_{21}}{x_{21}}(x_{a}-x_{1})-y\bigg{]} +z_{21}\bigg{[}z_{1}+\frac{z_{21}}{x_{21}}(x_{a}-x_{1})-z\bigg{]}=0;\]0  
\[x_{21}(x_{a}-x)+y_{21}\bigg{[}y_{1}+\frac{y_{21}}{x_{21}}(x_{a}-x_{1})-y\bigg{]} +z_{21}\bigg{[}z_{1}+\frac{z_{21}}{x_{21}}(x_{a}-x_{1})-z\bigg{]}=0;\]1  
\[x_{21}(x_{a}-x)+y_{21}\bigg{[}y_{1}+\frac{y_{21}}{x_{21}}(x_{a}-x_{1})-y\bigg{]} +z_{21}\bigg{[}z_{1}+\frac{z_{21}}{x_{21}}(x_{a}-x_{1})-z\bigg{]}=0;\]2  
After calculating the intensities using Formulas (9) and (17), we can find the projections on the coordinate axes.