Sage Embedding
Using the Sage Cell Server
This code snippet shows the essential PG code to embed a call to the Sage Cell Server from within a problem. Note that these are insertions, not a complete PG file. This code will have to be incorporated into the problem file on which you are working.
PG problem file  Explanation 

########################################################### ## ## pg initializations and regular WeBWorK code $a11 = random(2,3,1/2); $a12 = 1; $a21 = random(3,1,1/2); $a22 = non_zero_random(2,5,1/20); $A = Matrix([[$a11,$a12],[$a21,$a22]]); $A1 = Vector($a11,$a21); $x1 = non_zero_random(2,2,1/20); $x1ans = Compute("$x1"); $x2 = non_zero_random(2,2,1/10); $x2ans = Compute("$x2"); $x = Vector($x1,$x2); $b1 = $a11*$x1+$a12*$x2; $b2 = $a21*$x1+$a22*$x2; $b = Vector($b1,$b2); 

Next you add a section which contains the actual answer call and Sage code needed to implement the desired function.
Notice, the answer call appears near the top between a tag and a <script> tag. Multiple results will need to have multiple answer calls. The numerical value for the size of the "answer blank" is unimportant since this blank will eventually be overwritten by the Sage Cell.
Working Sage code will work verbatim except for a couple of notational changes caused by conflicting syntax between perl and sage. in particular, since "@" is used for tables in perl and for interacts in sage, one will need to replace "@" with "~~@". Further, perl uses \( and \) to delimit latex and "$" for variables while Sage uses "$' to delimit latex. Therefore, changing each of Sage's latex delimiters to the \( and \) format averts any conflict. An example of they usage is illustrated in the code fragment below. 

BEGIN_TEXT <div id="sagecell"> The solution x for Ax=b is given by x1=\{ ans_rule(15) \} and x2=\{ans_rule(15) \}. <script type="text/code"> b = matrix([[$b1],[$b2]]) bt = b.transpose() A=matrix([[$a11,$a12],[$a21,$a22]]) At =A.transpose() # Notice the correct exact answer is given by x = A\b # Finding when a vector b is in the span of other vectors in 2space <b>~~@interact</b> def _(x1=slider(3,3,1/20,1), x2=slider(3,3,1/20,1)): G = arrow((0,0),x1*At[0],rgbcolor=(0,0,1)) G += arrow(x1*At[0],x1*At[0]+x2*At[1],rgbcolor=(0,1,0)) G += arrow((0,0),($b1,$b2),rgbcolor=(1,0,0),width=5) G += text("A1",(x1*At[0][0]/2,x1*At[0][1]/2),fontsize=30,color='purple') G += text("A2",(x1*At[0][0]+x2*At[1][0]/2,x1*At[0][1]+x2*At[1][1]/2),fontsize=30,color='purple') G += text("b",($b1/2,$b2/2),fontsize=40,color='purple') G += point(x1*At[0],color='blue',pointsize=40) G += point(($b1,$b2),color='red',pointsize=30) G += point(x1*At[0]+x2*At[1],color='green',pointsize=40) G += point(($b1,$b2),color='red',pointsize=20) # Add fixed originals and dashed modified version of these show(G,frame=False) html('<input type=hidden size=15 name="\{ANS_NUM_TO_NAME(1)\}" id="\{ANS_NUM_TO_NAME(1)\}" value="%s">' %str(x1) ) html('<input type=hidden size=15 name="\{ANS_NUM_TO_NAME(2)\}" id="\{ANS_NUM_TO_NAME(2)\}" value="%s">' %str(x2) ) </script> </div> 
This snippet defines the applet
The
The
FlashApplets describes this interface from the flash applets point of view. 
################################### # Configure applet ################################### #data to set up the equation $applet>configuration(qq{<XML expr='$function' />}); # initial points $applet>initialState(qq{<XML> <pt xval='0' yval='0'/></XML>}); ################################### #insert applet into body ################################### TEXT( MODES(TeX=>'object code', HTML=>$applet>insertAll( debug=>0, includeAnswerBox=>1, reinitialize_button=>$permissionLevel>=10, ))); 
Now we configure the applet. The contents of The 
BEGIN_TEXT $PAR Drag the point to the inflection point of the given curve and press the submit button. END_TEXT 
The problem text section of the file is as we'd expect. 
NAMED_ANS('answerBox'=>$answer_point >with(tolType=>"absolute",tolerance=>.05) >cmp >withPostFilter(AnswerHints( sub { my ($correct,$student,$ans) = @_; return Vector($correct$student)>norm<.2 ; } => ["You're close. You need to position the dot more precisely.", replaceMessage=>1] ))); 
The answer checker grabs the answer from the default 
Problem Techniques Index
More on how to embed applets in WeBWorK Questions
 POD documentation: AppletObjects.pl.html
 PG macro: AppletObjects.pl