Q: Drawing lines, with arrowheads, at arbitrary angles with respect to the positive x-axis I am trying to get TikZ to draw two lines, with arrowheads, on the Cartesian plane.  I have drawn a square, with gray edges, and I want the tip of the arrowheads to be on an edge of the square.  In particular, I want to include the six commands following the comment 
"The following code is for placing arrowheads at the ends of the line segments."

TikZ is not compiling them. (This can be an illustrative example using the intersections package.)
\documentclass{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,backgrounds}


\begin{document}



\begin{tikzpicture}[outer sep=0pt,p/.style={circle, fill,inner sep=1.5pt}]

\draw[draw=gray!30,latex-latex] (-3.75,0) +(-0.25cm,0) -- (3.75,0) -- +(0.25cm,0) node[below right] {$x$};

\clip (-3.75,-3.75) rectangle (3.75,3.75);

\draw[gray,dashed,line width=0.1pt] (-3.75,3.75) -- (3.75,3.75);
\draw[gray,dashed,line width=0.1pt] (-3.75,-3.75) -- (3.75,-3.75);
\draw[gray,dashed,line width=0.1pt] (-3.75,-3.75) -- (-3.75,3.75);
\draw[gray,dashed,line width=0.1pt] (3.75,-3.75) -- (3.75,3.75);

\draw[draw=blue!30,-latex] (0,0) -- (142:5);
\draw[draw=blue!30,-latex] (0,0) -- (-38:5);
\draw[draw=green!50,-latex] (0,0) -- (52:5);
\draw[draw=green!50,-latex] (0,0) -- (-128:5);

\coordinate[p,label={[fill=white]below right:$O$}] (O) at (0,0);

\coordinate (A) at (0:1);
\coordinate (B) at (52:1);
\path pic[draw, angle radius=5mm,"$\phi$",angle eccentricity=1.25] {angle = A--O--B};

\coordinate (a) at (180:1);
\coordinate (b) at (142:1);
\path pic[draw, angle radius=5mm,"$\theta$",angle eccentricity=1.25] {angle = b--O--a};

\coordinate (P) at (142:1);
\coordinate (Q) at (52:1);

\coordinate (R) at ($(O)!4mm! -45:(P)$);
\draw (R) -- ($(O)!(R)!(P)$);
\draw (R) -- ($(O)!(R)!(Q)$);

%The following code makes the right-angle mark and "colors" the inside of it white.
\begin{scope}[on background layer]
\draw[draw=gray!30,latex-latex] (0,3.75) +(0,0.25cm) node[above right] {$y$} -- (0,-3.75) -- +(0,-0.25cm);
\filldraw[fill=white] (O.center) -- ($(O)!(R)!(P)$) -- (R) -- ($(O)!(R)!(Q)$) -- cycle;
\end{scope}


%The following code is for placing arrowheads at the ends of the line segments.
%\path[name intersections={of=(-3.75,3.75) -- (3.75,3.75) and (0,0) -- (52:5), by=intersection-1}];
%\path[name intersections={of=(3.75,3.75) -- (3.75,-3.75) and (0,0) -- (-38:5), by=intersection-2}];
%\path[name intersections={of=(-3.75,-3.75) -- (3.75,-3.75) and (0,0) -- (-128:5), by=intersection-3}];
%\path[name intersections={of=(-3.75,3.75) -- (-3.75,-3.75) and (0,0) -- (142:5), by=intersection-4}];

%\draw[draw=green!50,latex-latex] (intersection-1) -- (intersection-3);
%\draw[draw=blue!30,latex-latex] (intersection-2) -- (intersection-4);

\end{tikzpicture}

\end{document}


A: Here's a Metapost effort for comparison.  The idea is that you set the angle phi and everything else adjusts automatically.

prologues := 3;
outputtemplate := "%j%c.eps";

beginfig(1);
u := 1cm;
% defime the paths we need and a value for phi..
path xx, yy, box, ray;
xx = (left--right) scaled 4u;
yy = xx rotated 90;
box = unitsquare shifted -(1/2,1/2) scaled 7u;
ray = origin -- right scaled 7u;
numeric phi;
phi = 57.3; 

% first draw axes and box
drawoptions(withcolor .7 white);
drawdblarrow xx;
drawdblarrow yy;
draw box dashed evenly;
drawoptions();

% now draw the angle marks so they are underneath the main lines
% angle marks assume 0 < phi < 90...
path angle_mark[]; 
angle_mark1 = (1u,0) {up} .. (1u,0) rotated phi;
angle_mark2 = (-u,0) {up} .. (-u,0) rotated (phi-90);
angle_mark3 = unitsquare scaled 3/8 u rotated phi;
draw angle_mark1;
draw angle_mark2;
unfill angle_mark3; 
draw angle_mark3;

% now the main lines
for i=0 upto 3:
  drawarrow ray rotated (phi+90i) cutafter box withcolor .42[if odd(i): blue else: red fi, white];
endfor

% finally the labels
label(btex $\phi$   etex, point 1/2 of angle_mark1 scaled 1.23);
label(btex $\theta$ etex, point 1/2 of angle_mark2 scaled 1.23);
label.rt (btex $x$ etex, point 1 of xx);
label.top(btex $y$ etex, point 1 of yy);
fill fullcircle scaled dotlabeldiam;
picture O;
O = thelabel(btex $O$ etex, (u/2,0) rotated (180+45+phi));
unfill bbox O; draw O;

endfig;
end.


A: Here is a shorter method.
\documentclass{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,backgrounds}


\begin{document}



\begin{tikzpicture}[outer sep=0pt,p/.style={circle, fill,inner sep=1.5pt}]

\draw[draw=gray!30,latex-latex] (-3.75,0) +(-0.8cm,0) -- (3.75,0) -- +(0.25cm,0) node[below right] {$x$};

\coordinate[p,label={[fill=white]below right:$O$}] (O) at (0,0);

\draw[draw=blue!30,-latex] (0,0) -- (142:5)coordinate (aa);
\draw[draw=blue!30,-latex] (0,0) -- (-38:4.5)coordinate (cc);
\draw[draw=green!50,-latex] (0,0) -- (52:4.5)coordinate (dd);
\draw[draw=green!50,-latex] (0,0) -- (-128:4.5)coordinate (bb);

\draw[gray,dashed,line width=0.1pt] (aa|-bb) rectangle (cc|-dd);

\coordinate[p,label={[fill=white]below right:$O$}] (O) at (0,0);

\coordinate (A) at (0:1);
\coordinate (B) at (52:1);
\path pic[draw, angle radius=5mm,"$\phi$",angle eccentricity=1.25] {angle = A--O--B};

\coordinate (a) at (180:1);
\coordinate (b) at (142:1);
\path pic[draw, angle radius=5mm,"$\theta$",angle eccentricity=1.25] {angle = b--O--a};

\coordinate (P) at (142:1);
\coordinate (Q) at (52:1);

\coordinate (R) at ($(O)!4mm! -45:(P)$);
\draw (R) -- ($(O)!(R)!(P)$);
\draw (R) -- ($(O)!(R)!(Q)$);

%The following code makes the right-angle mark and "colors" the inside of it white.
\begin{scope}[on background layer]
\draw[draw=gray!30,latex-latex] (0,3.75) +(0,0.25cm) node[above right] {$y$} -- (0,-3.75) -- +(0,-0.25cm);
\filldraw[fill=white] (O.center) -- ($(O)!(R)!(P)$) -- (R) -- ($(O)!(R)!(Q)$) -- cycle;
\end{scope}

\end{tikzpicture}

\end{document}


No clipping, no drawing the rectangle piece by piece, no intersection business. Just define coordinates aa, bb etc, and then use
\draw[gray,dashed,line width=0.1pt] (aa|-bb) rectangle (cc|-dd);

to draw the rectangle. I have adjusted the length of those lines from 5 to 4.5 BTW.
Since I have some confusion still in my mind, you can draw other lines too like
\draw[olive,latex-latex] (aa) -- (bb);
\draw[olive,latex-latex] (aa) -- (dd);


Edit
To keep \clip and find intersections, you have to name the path when it is being drawn. Check the following code.
\documentclass[10pt]{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,quotes,backgrounds,intersections}
%% come back here

\begin{document}


\begin{tikzpicture}[outer sep=0pt,p/.style={circle, fill,inner sep=1.5pt}]

\draw[draw=gray!30,latex-latex] (-3.75,0) +(-0.25cm,0) -- (3.75,0) -- +(0.25cm,0) node[below right] {$x$};

\clip (-3.75,-3.75) rectangle (3.75,3.75);

\draw[gray,dashed,line width=0.1pt,name path =A] (-3.75,3.75) -- (3.75,3.75);
\draw[gray,dashed,line width=0.1pt,name path = E] (-3.75,-3.75) -- (3.75,-3.75);
\draw[gray,dashed,line width=0.1pt,name path =G] (-3.75,-3.75) -- (-3.75,3.75);
\draw[gray,dashed,line width=0.1pt,name path=C] (3.75,-3.75) -- (3.75,3.75);

\draw[draw=blue!30,-latex,name path = H] (0,0) -- (142:5);
\draw[draw=blue!30,-latex,name path = D] (0,0) -- (-38:5);
\draw[draw=green!50,-latex,name path=B] (0,0) -- (52:5);
\draw[draw=green!50,-latex,name path =F] (0,0) -- (-128:5);

\coordinate[p,label={[fill=white]below right:$O$}] (O) at (0,0);

\coordinate (A) at (0:1);
\coordinate (B) at (52:1);
\path pic[draw, angle radius=5mm,"$\phi$",angle eccentricity=1.25] {angle = A--O--B};

\coordinate (a) at (180:1);
\coordinate (b) at (142:1);
\path pic[draw, angle radius=5mm,"$\theta$",angle eccentricity=1.25] {angle = b--O--a};

\coordinate (P) at (142:1);
\coordinate (Q) at (52:1);

\coordinate (R) at ($(O)!4mm! -45:(P)$);
\draw (R) -- ($(O)!(R)!(P)$);
\draw (R) -- ($(O)!(R)!(Q)$);

\begin{scope}[on background layer]
\draw[draw=gray!30,latex-latex] (0,3.75) +(0,0.25cm) node[above right] {$y$} -- (0,-3.75) -- +(0,-0.25cm);
\filldraw[fill=white] (O.center) -- ($(O)!(R)!(P)$) -- (R) -- ($(O)!(R)!(Q)$) -- cycle;
\end{scope}

    %The following code is for placing arrowheads at the ends of the line segments.
    \path[name intersections={of= A and B, by=aa}];
    \path[name intersections={of=C  and D, by=bb}];
    \path[name intersections={of= E and F, by=cc}];
    \path[name intersections={of= G and H, by=dd}];
    \draw[draw=green!30,latex-latex] (O) -- (aa);
    \draw[draw=blue!30,latex-latex] (O) -- (bb);
    \draw[draw=green!50,latex-latex] (O) -- (cc);
    \draw[draw=blue!30,latex-latex] (O) -- (dd);



\end{tikzpicture}

\end{document}


