Practical Linear Algebra: A Geometry Toolbox, Third edition
Gerald Farin & Dianne Hansford
CRC Press, Taylor & Francis Group, An A K Peters Book, 2013

Determinants 3x3 matrices
Three vectors a, b, c span a parallelepiped ( a skew box). It volume is given by the determinant of a, b, c. If it is nonzero, the vectors are linearly independent.

aa = {1,0,0};  (*  green *)
bb= {0,1,0};  (*  green *)
cc= {0,1,0};  (*  green*)

(* viewing extents *)
range=2*Max[Norm[aa],Norm[bb],Norm[cc]];

arrowaa={Green,Arrowheads[Large],Arrow[{{0,0,0},aa}]};
arrowbb ={Green,Arrowheads[Large],Arrow[{{0,0,0},bb}]};
arrowcc ={Green,Arrowheads[Large],Arrow[{{0,0,0},cc}]};
zz={0,0,0};

Manipulate[
rst={r,s,t};
Graphics3D[{ Arrowheads[.05],Arrow[Tube[{{0,0,0},rst},.015]],Thick,arrowaa, arrowbb, arrowcc,LightRed,Opacity[0.8],
Polygon[{zz,aa,aa+bb,bb}],
Polygon[{zz,aa,aa+rst,rst}],
Polygon[{zz,bb,bb+rst,rst}],
Polygon[{rst+zz,rst+aa,rst+aa+bb,rst+bb}],
Polygon[{bb+zz,bb+aa,bb+aa+rst,bb+rst}],
Polygon[{aa+zz,aa+bb,aa+bb+rst,aa+rst}]
},Boxed->False,PlotRange->range,PlotLabel->Style[Framed[Det[{aa,bb,rst}]],20]],{{r,0,"e_1 dir:  r"},-1,1},{{s,0,"e_2 dir:  s"},-1,1},{{t,1,"e_3 dir:  t"},-1,1},
FrameLabel->{"Tube arrow defined by (r,s,t)"," ","Determinant: area of the parallelepiped"," "}]