Csdrhrt

I am often use coordinates of points to draw figure in geometry. I know that, we can add, minus coordinates of points, example

begin{tikzpicture}

tkzDefPoints{0/0/C',3/0/D',1/1/B'}

coordinate (A') at ($(B')+(D')-(C')$);

 end{tikzpicture}

If I have two points A(1,2,3) and B(4,5,6), how can I define vector AB as (B)-(A)?

If you use the coordinates only for drawing, simply define each components of points as variable and then define coordinate points using them. For example:

documentclass[margin=3.14159mm]{standalone}

usepackage{tikz,tikz-3dplot}



begin{document}

tdplotsetmaincoords{60}{125}

begin{tikzpicture}

    [scale=0.9,

    tdplot_main_coords,

    axis/.style={-latex,thick},

    vector/.style={-stealth,red,very thick},

    vector guide/.style={dashed,thick}]



%standard tikz coordinate definition using x, y, z coords

% A(2,4,3), B(3,-1,4)

defAx{2}

defAy{4}

defAz{3}

defBx{-1}

defBy{3}

defBz{4}

coordinate (O) at (0,0,0);

coordinate (A) at (Ax,Ay,Az);

coordinate (B) at (Bx,By,Bz);

%draw axes

draw[axis] (0,0,0) -- (4,0,0) node[anchor=north east]{$x$};

draw[axis] (0,0,0) -- (0,4,0) node[anchor=north west]{$y$};

draw[axis] (0,0,0) -- (0,0,5) node[anchor=south]{$z$};

%Dot at point

fill [blue] (A) circle (2pt);

fill [blue] (B) circle (2pt);

%draw a vector from O to A and O to B

draw[vector guide] (O)node[left=1mm]{} -- (A)node[above=-1mm,right]{$P_1(Ax,Ay,Az)$};

draw[vector guide] (O) -- (B)node[above=-1mm,right]{$P_2(Bx,By,Bz)$};



%draw vector D=AB

draw[vector] (A) -- (B)node[midway,above,sloped]{$mathbf{D}$};

end{tikzpicture}

end{document}

SUPPLEMENT

With the permission of the answerer, I (Steven B Segletes) show here how the listofitems package can be used to streamline the syntax and maybe provide more readability. With it, I can create the arrays by reading a list, with the syntax readlistA{2,4,3}. Then, the expression A will spit back the array 2,4,3, which is sufficient for use in the present MWE. However, the individual components are also accessible as A[1], A[2], and A[3], which can be used for various calculations, as required.

documentclass[margin=3.14159mm]{standalone}

usepackage{tikz,tikz-3dplot,listofitems}



begin{document}

tdplotsetmaincoords{60}{125}

begin{tikzpicture}

    [scale=0.9,

    tdplot_main_coords,

    axis/.style={-latex,thick},

    vector/.style={-stealth,red,very thick},

    vector guide/.style={dashed,thick}]



%standard tikz coordinate definition using x, y, z coords

% A(2,4,3), B(3,-1,4)

readlistA{2,4,3}

readlistB{-1,3,4}

coordinate (O) at (0,0,0);

coordinate (A) at (A);

coordinate (B) at (B);

%draw axes

draw[axis] (0,0,0) -- (4,0,0) node[anchor=north east]{$x$};

draw[axis] (0,0,0) -- (0,4,0) node[anchor=north west]{$y$};

draw[axis] (0,0,0) -- (0,0,5) node[anchor=south]{$z$};

%Dot at point

fill [blue] (A) circle (2pt);

fill [blue] (B) circle (2pt);

%draw a vector from O to A and O to B

draw[vector guide] (O)node[left=1mm]{} -- (A)node[above=-1mm,right]{$P_1(A)$};

draw[vector guide] (O) -- (B)node[above=-1mm,right]{$P_2(B)$};



%draw vector D=AB

draw[vector] (A) -- (B)node[midway,above,sloped]{$mathbf{D}$};

end{tikzpicture}

end{document}

Just for fun, I wrote routines for 3D vector addition, subtraction, cross product and dot product (scalar treated as a 1D vector). I was trying to actually parse expressions of the form A+B but eventually gave up.

documentclass{article}

usepackage{listofitems}

usepackage{pgfmath}

usepackage{amsmath}



makeatletter

newcommand{@vecargs}{}% reserve global names



newcommand{vecadd}{}

newcommand{vecsub}{}

newcommand{vecdot}{}

newcommand{veccross}{}

newcommand{vecparse}{}



defvecadd#1#2#3% #1 = #2 + #3

{bgroup% local definitions

  pgfmathsetmacro{@x}{#2[1]+#3[1]}%

  pgfmathsetmacro{@y}{#2[2]+#3[2]}%

  pgfmathsetmacro{@z}{#2[3]+#3[3]}%

  xdef@vecargs{@x,@y,@z}%

egroup

readlist#1{@vecargs}}



defvecsub#1#2#3% #1 = #2 - #3

{bgroup% local definitions

  pgfmathsetmacro{@x}{#2[1]-#3[1]}%

  pgfmathsetmacro{@y}{#2[2]-#3[2]}%

  pgfmathsetmacro{@z}{#2[3]-#3[3]}%

  xdef@vecargs{@x,@y,@z}%

egroup

readlist#1{@vecargs}}



defvecdot#1#2#3% #1 = #2 cdot #3

{pgfmathsetmacro{@vecargs}{#2[1]*#3[1] + #2[2]*#3[2] + #3[3]*#3[3]}%

readlist#1{@vecargs}}



defveccross#1#2#3% #1 = #2 times #3

{bgroup% local definitions

  pgfmathsetmacro{@x}{#2[2]*#3[3] - #2[3]*#3[2]}%

  pgfmathsetmacro{@y}{#2[3]*#3[1] - #2[1]*#3[3]}%

  pgfmathsetmacro{@z}{#2[1]*#3[2] - #2[2]*#3[1]}%

  xdef@vecargs{@x,@y,@z}%

egroup

readlist#1{@vecargs}}

makeatother



begin{document}

readlistA{1,2,3}

readlistB{4,5,6}



vecaddCAB

C



vecsubCAB

C



vecdotCAB

C



veccrossCAB

C

end{document}

SUPPLEMENT

I hope John doesn't mind me (Steven B Segletes) adding his sought-after parser to the code. This allows input of the form vecparseC{A+B}, vecparseC{A - B}, vecparseC{A .B}, and vecparseC{A xB} (extra spaces of no consequence).

Support added not only for vecparseC{A xB}, but also vecparseC{A x(3,5,6)}, vecparseC{(3,5,6)xB} and vecparseC{(1,1,1)x(1,2,3)}.

documentclass{article}

usepackage{listofitems}

usepackage{pgfmath}

usepackage{amsmath}



makeatletter

newcommand{@vecargs}{}% reserve global names



newcommand{vecadd}{}

newcommand{vecsub}{}

newcommand{vecdot}{}

newcommand{veccross}{}

newcommand{vecparse}{}



defvecadd#1#2#3% #1 = #2 + #3

{bgroup% local definitions

  pgfmathsetmacro{@x}{#2[1]+#3[1]}%

  pgfmathsetmacro{@y}{#2[2]+#3[2]}%

  pgfmathsetmacro{@z}{#2[3]+#3[3]}%

  xdef@vecargs{@x,@y,@z}%

egroup

setsepchar{,}%

readlist#1{@vecargs}}



defvecsub#1#2#3% #1 = #2 - #3

{bgroup% local definitions

  pgfmathsetmacro{@x}{#2[1]-#3[1]}%

  pgfmathsetmacro{@y}{#2[2]-#3[2]}%

  pgfmathsetmacro{@z}{#2[3]-#3[3]}%

  xdef@vecargs{@x,@y,@z}%

egroup

setsepchar{,}%

readlist#1{@vecargs}}



defvecdot#1#2#3% #1 = #2 cdot #3

{pgfmathsetmacro{@vecargs}{#2[1]*#3[1] + #2[2]*#3[2] + #3[3]*#3[3]}%

setsepchar{,}%

readlist#1{@vecargs}}



defveccross#1#2#3% #1 = #2 times #3

{bgroup% local definitions

  pgfmathsetmacro{@x}{#2[2]*#3[3] - #2[3]*#3[2]}%

  pgfmathsetmacro{@y}{#2[3]*#3[1] - #2[1]*#3[3]}%

  pgfmathsetmacro{@z}{#2[1]*#3[2] - #2[2]*#3[1]}%

  xdef@vecargs{@x,@y,@z}%

egroup

setsepchar{,}%

readlist#1{@vecargs}}



defvecparse#1#2{%

  setsepchar{+||-||x||./(||)}%

  readlist*@findop{#2}%

  ifnumlistlen@findop[1]=1relax

    itemtomacro@findop[1]tmpA

  else

    itemtomacro@findop[1,2]tmpF

    setsepchar{,}%

    readlisttmpE{tmpF}%

    deftmpA{tmpE}%

  fi

  ifnumlistlen@findop[2]=1relax

    itemtomacro@findop[2]tmpB

  else

    itemtomacro@findop[2,2]tmpD

    setsepchar{,}%

    readlisttmpC{tmpD}%

    deftmpB{tmpC}%

  fi

  if+@findopsep[1]relax

    deftmp{vecadd#1}%

  elseif-@findopsep[1]relax

    deftmp{vecsub#1}%

  elseif.@findopsep[1]relax

    deftmp{vecdot#1}%

  elseif x@findopsep[1]relax

    deftmp{veccross#1}%

  fifififi

  expandafterexpandafterexpandaftertmpexpandaftertmpAtmpB

}

makeatother



begin{document}

readlistA{1,2,3}

readlistB{4,5,6}



vecaddCAB

C



VP:vecparseC{A+B}

C



vecsubCAB

C



VP:vecparseC{A - B}

C



vecdotCAB

C



VP:vecparseC{A .B}

C 



veccrossCAB

C



VP:vecparseC{A xB}

C



VP:vecparseC{A x(3,5,6)}

C



VP:vecparseC{(3,5,6)xB}

C



VP:vecparseC{(1,1,1)x(1,2,3)}

C



end{document}