CliffordBasic calculations for Figure 1.20 reflection (reflectionFig1.eps), but not the figure itself.  Also has mmacell output for the input and output cells for this calculation.

<< CliffordBasic`;
$SetSignature={2,0};
Import["https://raw.githubusercontent.com/jkuczm/MathematicaCellsToTeX/master/NoInstall.m"]

ClearAll[u, x, uu, invu, i, o, proj, rej, ux, uxu]
u = 4 e[1] + 2 e[2];
x = 3 e[1] + 3 e[2];
uu = InnerProduct[u, u];
invu = u/uu;
i = InnerProduct[x, u] ;
o = OuterProduct[x,u];
proj = i invu // N//Simplify
rej = GeometricProduct[ o, invu ] //N//Simplify
ux = GeometricProduct[ u, x]
uxu = GeometricProduct[ ux, invu ] // N// Simplify

Construct mmaCell’s

CellToTeX[Cell[BoxData[{RowBox[{"ClearAll","[",RowBox[{"u",","," ","x",","," ","uu",","," ","invu",","," ","i",","," ","o",","," ","proj",","," ","rej",","," ","ux",","," ","uxu"}],"]"}],"
",RowBox[{RowBox[{"u"," ","="," ",RowBox[{RowBox[{"4"," ",RowBox[{"e","[","1","]"}]}]," ","+"," ",RowBox[{"2"," ",RowBox[{"e","[","2","]"}]}]}]}],";"}],"
",RowBox[{RowBox[{"x"," ","="," ",RowBox[{RowBox[{"3"," ",RowBox[{"e","[","1","]"}]}]," ","+"," ",RowBox[{"3"," ",RowBox[{"e","[","2","]"}]}]}]}],";"}],"
",RowBox[{RowBox[{"uu"," ","="," ",RowBox[{"InnerProduct","[",RowBox[{"u",","," ","u"}],"]"}]}],";"}],"
",RowBox[{RowBox[{"invu"," ","="," ",RowBox[{"u","/","uu"}]}],";"}],"
",RowBox[{RowBox[{"i"," ","="," ",RowBox[{"InnerProduct","[",RowBox[{"x",","," ","u"}],"]"}]}]," ",";"}],"
",RowBox[{RowBox[{"o"," ","="," ",RowBox[{"OuterProduct","[",RowBox[{"x",",","u"}],"]"}]}],";"}],"
",RowBox[{"proj"," ","="," ",RowBox[{RowBox[{RowBox[{"i"," ","invu"}]," ","//"," ","N"}],"//","Simplify"}]}],"
",RowBox[{"rej"," ","="," ",RowBox[{RowBox[{RowBox[{"GeometricProduct","["," ",RowBox[{"o",","," ","invu"}]," ","]"}]," ","//","N"}],"//","Simplify"}]}],"
",RowBox[{"ux"," ","="," ",RowBox[{"GeometricProduct","["," ",RowBox[{"u",","," ","x"}],"]"}]}],"
",RowBox[{"uxu"," ","="," ",RowBox[{RowBox[{RowBox[{"GeometricProduct","["," ",RowBox[{"ux",","," ","invu"}]," ","]"}]," ","//"," ","N"}],"//"," ","Simplify"}]}]}],"Input",CellChangeTimes->{{3.727606036795764*^9,3.7276064189353943`*^9},{3.727628105293309*^9,3.727628124365094*^9},{3.727628156937948*^9,3.727628200856783*^9},{3.728471096384516*^9,3.728471116270248*^9}}]]

CellToTeX[Cell[BoxData[RowBox[{RowBox[{"3.6`"," ",RowBox[{"e","[","1","]"}]}],"+",RowBox[{"1.8`"," ",RowBox[{"e","[","2","]"}]}]}]],"Output",CellChangeTimes->{{3.72760626190495*^9,3.7276062881187277`*^9},{3.727606319147373*^9,3.727606340341743*^9},{3.727606375872643*^9,3.7276064201066923`*^9},{3.727627952983482*^9,3.7276279779058447`*^9},{3.72762813293769*^9,3.7276281621674547`*^9},3.728471121265671*^9,3.728471220570485*^9}]]

CellToTeX[Cell[BoxData[RowBox[{RowBox[{RowBox[{"-","0.6`"}]," ",RowBox[{"e","[","1","]"}]}],"+",RowBox[{"1.2`"," ",RowBox[{"e","[","2","]"}]}]}]],"Output",CellChangeTimes->{{3.72760626190495*^9,3.7276062881187277`*^9},{3.727606319147373*^9,3.727606340341743*^9},{3.727606375872643*^9,3.7276064201066923`*^9},{3.727627952983482*^9,3.7276279779058447`*^9},{3.72762813293769*^9,3.7276281621674547`*^9},3.728471121265671*^9,3.7284712205815067`*^9}]]

CellToTeX[Cell[BoxData[RowBox[{"18","+",RowBox[{"6"," ",RowBox[{"e","[",RowBox[{"1",",","2"}],"]"}]}]}]],"Output",CellChangeTimes->{{3.72760626190495*^9,3.7276062881187277`*^9},{3.727606319147373*^9,3.727606340341743*^9},{3.727606375872643*^9,3.7276064201066923`*^9},{3.727627952983482*^9,3.7276279779058447`*^9},{3.72762813293769*^9,3.7276281621674547`*^9},3.728471121265671*^9,3.728471220589456*^9}]]

CellToTeX[Cell[BoxData[RowBox[{RowBox[{"4.2`"," ",RowBox[{"e","[","1","]"}]}],"+",RowBox[{"0.6`"," ",RowBox[{"e","[","2","]"}]}]}]],"Output",CellChangeTimes->{{3.72760626190495*^9,3.7276062881187277`*^9},{3.727606319147373*^9,3.727606340341743*^9},{3.727606375872643*^9,3.7276064201066923`*^9},{3.727627952983482*^9,3.7276279779058447`*^9},{3.72762813293769*^9,3.7276281621674547`*^9},3.728471121265671*^9,3.728471220597274*^9}]]

Manual cell construction for first mmaCell experiments.

testCell=Cell[BoxData[MakeBoxes[u = 4 e[1] + 2 e[2];
x = 3 e[1] + 3 e[2];
uu = InnerProduct[u, u];
invu = u/uu;
i = InnerProduct[x, u] ;
o = OuterProduct[x,u];
proj = i invu // N//Simplify;
rej = GeometricProduct[ o, invu ] //N//Simplify;
ux = GeometricProduct[ u, x];
uxu = GeometricProduct[ ux, invu ] // N// Simplify]],"Input"];

CellToTeX[testCell]