☰

使用 TikZ 计算叉积

tag-icon • tag-icon tikz-pgf
使用 TikZ 计算叉积

有没有办法使用 TikZ 来计算两个向量的叉积？

这是我的代码：

\begin{tikzpicture}[scale=2]
  \coordinate (ey) at (1,0,0);
  \coordinate (ez) at (0,1,0);
  \coordinate (ex) at (0,0,1);
  
  \begin{scope}[x={(ex)},y={(ey)},z={(ez)}]
    \def\xM{1.2}
    \def\yM{1.6}
    \def\zM{1.8}
    \def\ech{2}  
    \coordinate (O) at (0,0,0);
    \coordinate (X) at (\ech,0,0);
    \coordinate (Y) at (0,\ech,0);
    \coordinate (Z) at (0,0,\ech);
    \coordinate (Pz) at (0,0,\zM);
    \coordinate (M) at (\xM,\yM,\zM);
    \coordinate (H) at (\xM,\yM,0); 
    \coordinate (V) at ($0.4*(H)+0.48*(Pz)$);
    \coordinate (W) at ($0.4*(H)$);
    \coordinate (Vz) at ($0.48*(Pz)$);
    \coordinate (U) at ($0.33*(Pz)$);
   
    \draw[->,thick] (0,0,0) -- (1,0,0) node[above left] {$\vec{\imath}$};
    \draw[->,thick] (0,0,0) -- (0,1,0) node[below] {$\vec{\jmath}$};
    \draw[->,thick] (0,0,0) -- (0,0,1) node[left] {$\vec{k}$};
      \draw (O) -- (X);
        \draw (O) -- (Y);
        \draw (O) -- (Z);
        \draw (O) -- (H);
        \draw[very thick,->,blue] (O) -- (V) node[above right] {$\vec{v}$};
        \draw[very thick,->,blue] (O) -- (W) node[below] {$\vec{w}$};
        \draw[very thick,->,blue] (O) -- (U) node[left] {$\vec{u}$};
        \draw (M) -- (H);
        \draw (M) -- (Pz);
        \draw[ultra thin] (V) -- (Vz);
        \draw[ultra thin] (V) -- (W);
    
        \path (O) node[left] {$O$};
        \pgfmathsetmacro\angletheta{atan(\yM/\xM)} % rotation angle for OHM plane (theta)
    \draw pic [rotate around z=\angletheta,canvas is xz plane at y=0,% draws in the OHM plane
               draw,angle radius=0.75cm,
               "$\theta$",              % angle label (requires quotes library)
                angle eccentricity=1.4   % angle position
               ] {angle = V--O--Z};
    \draw[ultra thin] pic [draw,angle radius=0.2cm] {right angle = V--Vz--O};
    \draw pic [canvas is xy plane at z=0,% draws in the xy plane
               draw,->,angle radius=0.75cm,
               "$\varphi$",               % angle label (requires quotes library)
               angle eccentricity=1.8    % angle position
              ] {angle = X--O--H};
    \end{scope}
\end{tikzpicture}

结果如下： 在此处输入图片描述

所以，现在我想在平面 Oxy（以原点 O）上绘制矢量 n，它是 ux v 的叉积。

我怎样才能做到这一点？

答案1

在此处输入图片描述

使用tikz-3dplot包。从您的示例中，我了解到v上的投影Oxy是w。一切都取决于两个角度，phi 和 theta，以及u和的范数w。它们是在开始时定义的。

代码

\documentclass[11pt, border=10pt]{standalone}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{math, calc}
\begin{document}

\tikzmath{%
  real \f, \t, \uNorm, \wNorm, \bound;
  \f = 40; % phi angle
  \t = 35; % theta angle
  \uNorm = .6;
  \wNorm = 1.3;
  \bound = 1.5;
}
\tikzset{%
  vec/.style={#1, thick, ->}
}

\tdplotsetmaincoords{75}{120}
\begin{tikzpicture}[every node/.style={scale=.7}, scale=2]
  \begin{scope}[tdplot_main_coords]
    \draw[thin] (0, 0, 0) -- (\bound, 0, 0) node[pos=1.1] {$x$};
    \draw[thin] (0, 0, 0) -- (0, \bound, 0) node[pos=1.1] {$y$};
    \draw[thin] (0, 0, 0) -- (0, 0, \bound) node[pos=1.1] {$z$};

    \draw[vec=red] (0, 0, 0) -- (1, 0, 0) node[below] {$i$};
    \draw[vec=red] (0, 0, 0) -- (0, 1, 0) node[above] {$j$};
    \draw[vec=red] (0, 0, 0) -- (0, 0, 1) node[left] {$k$};
    
    \draw[vec=blue] (0, 0, 0) -- (0, 0, \uNorm)
    coordinate (u) node[left] {$u$};
    \begin{scope}[rotate around z=\f, canvas is zx plane at y=0]
      \draw[very thin, gray!50] (0, 0) rectangle (\bound, 1.4*\bound);
      \draw[vec=blue] (0, 0) -- (0, \wNorm) coordinate (w) node[left] {$w$};
      \draw[vec=blue] (0, 0) -- ({\wNorm*tan(\t)}, \wNorm)
      coordinate (v) node[right] {$v$};
      \draw[red, dashed, very thin] (w) -- (v)
      let \p1 = (v) in -- (\x1, 0);
    \end{scope}

    %\draw[red, dashed] (v) -- (w);
    \draw[vec=blue, rotate around z=\f]
    (0, 0, 0) -- (0, -\uNorm*\wNorm, 0)  node[above] {$u\times v$};
  \end{scope}

\end{tikzpicture}
\end{document}

相关内容