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I'm trying really hard to draw the plane in the image using TikZ with one arrow sticking out of it and two arrows along the plane. Any help would be sincerely appreciated! Thank you very much. :) Image of the plane

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    Please, be so kind and show what you try so far! – Zarko Nov 29 '16 at 20:18
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Your sketch of vector product is very simple and can be drawn with plain TikZ:

\documentclass[tikz, border=3mm]{standalone}
\usetikzlibrary{positioning}

\begin{document}
    \begin{tikzpicture}
% plane
\draw (0,0) -- ++ (45:2.2) -- ++ (3.3,0) -- ++ (225:2.2) -- cycle;
\draw[very thick,->] (22.5:1.4) -- ++ (0,2) node[left] {$a\times b$} ;
\draw[->]   (2,0.5) -- ++ (45:1) node[above right] {$a$};
\draw[->]   (2,0.5) -- ++ (1,0)  node[right] {$b$};
    \end{tikzpicture}
\end{document}

Note: in my school we usually drew vector in bottom left corner ...

enter image description here

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