How do I use newcommand to print a matrix everytime I input 6 values?

Question

I'm relatively new to LaTeX and I still don't really know how to use the newcommand feature at all (new to stackexchange as well). I've been trying to find out how to print something like this

I've tried various ways and so far the code with the least number of errors is this:
newcommand{xmatrix}[6]{
begin{vmatrix}
    $i & j & k$
    {u1} & {u2} & {u3}
    {v1} & {v2} & {v3}
end{vmatrix}}

I'm pretty sure I'm just using the newcommand part wrong but I want to be able to print the unit vectors in bold in the first row and the next 6 numbers are input. Thank you for any advice!

campa · Accepted Answer

You want to define a macro with six arguments: those six arguments are represented by #1, ..., #6 in the macro code. Since vmatrix is always in math mode there is no need to use $ dollar signs within it. You could therefore do something like this:
documentclass{article}

usepackage{amsmath} % for vmatrix

newcommand{xmatrix}[6]{%
begin{vmatrix}
    mathbf{i} & mathbf{j} & mathbf{k} 
    #1 & #2 & #3
    #4 & #5 & #6
end{vmatrix}}

begin{document}

[
xmatrix{u1}{u2}{u3}{v1}{v2}{v3}
]

end{document}

However, I find a macro taking six arguments a little bit unpleasant and prone to errors. I propose an alternative definition with two arguments, each of them being a comma-separated list.
documentclass{article}

usepackage{amsmath}

makeatletter
newcommand*{xmatrix}[2]{{%
   def@xmatrix##1,##2,##3,##4relax{##1 & ##2 & ##3}%
   begin{vmatrix}
    mathbf{i} & mathbf{j} & mathbf{k} 
    @xmatrix#1,,,relax
    @xmatrix#2,,,relax
   end{vmatrix}%
}}
makeatother

begin{document}

[
xmatrix{u1,u2,u3}{v1,v2,v3}
quad
xmatrix{u1}{,v2,v3} % just as example
]

end{document}

egreg · Answer

You can do much better than defining a macro with six arguments. I propose just two arguments, representing the coefficients of the vectors as comma separated lists.
We can also define a variant that actually computes the cross product.
documentclass{article}
usepackage{amsmath,xparse}

ExplSyntaxOn

NewDocumentCommand{crossproduct}{s m m}
 {% #1 = optional *
  % #2 = first vector as comma list
  % #3 = second vector as comma list
  IfBooleanTF{#1}
   {% *-variant, compute the product
    msun_crossproduct_compute:nn { #2 } { #3 }
   }
   {% normal, typeset in determinant form
    msun_crossproduct_symbolic:nn { #2 } { #3 }
   }
 }

fp_new:N l_msun_crossproduct_i_fp
fp_new:N l_msun_crossproduct_j_fp
fp_new:N l_msun_crossproduct_k_fp

cs_new_protected:Nn msun_crossproduct_compute:nn
 {
  fp_set:Nn l_msun_crossproduct_i_fp
   {
    clist_item:nn { #1 } { 2 } * clist_item:nn { #2 } { 3 }
    -
    clist_item:nn { #1 } { 3 } * clist_item:nn { #2 } { 2 }
   }
  fp_set:Nn l_msun_crossproduct_j_fp
   {
    clist_item:nn { #1 } { 3 } * clist_item:nn { #2 } { 1 }
    -
    clist_item:nn { #1 } { 1 } * clist_item:nn { #2 } { 3 }
   }
  fp_set:Nn l_msun_crossproduct_k_fp
   {
    clist_item:nn { #1 } { 1 } * clist_item:nn { #2 } { 2 }
    -
    clist_item:nn { #1 } { 2 } * clist_item:nn { #2 } { 1 }
   }
  fp_use:N l_msun_crossproduct_i_fp mathbf{i}
  fp_compare:nF { l_msun_crossproduct_j_fp < c_zero_fp } { + }
  fp_use:N l_msun_crossproduct_j_fp mathbf{j}
  fp_compare:nF { l_msun_crossproduct_k_fp < c_zero_fp } { + }
  fp_use:N l_msun_crossproduct_k_fp mathbf{k}
 }

cs_new_protected:Nn msun_crossproduct_symbolic:nn
 {
  begin{vmatrix}
  mathbf{i} & mathbf{j} & mathbf{k}
  
  clist_item:nn { #1 } { 1 } & clist_item:nn { #1 } { 2 } & clist_item:nn { #1 } { 3 }
  
  clist_item:nn { #2 } { 1 } & clist_item:nn { #2 } { 2 } & clist_item:nn { #2 } { 3 }
  end{vmatrix}
 }

ExplSyntaxOff

begin{document}

[
crossproduct{0,1,-2}{3,0,-4}=
crossproduct*{0,1,-2}{3,0,-4}
]

end{document}

How do I use newcommand to print a matrix everytime I input 6 values?

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