Math 236  •  24 April 2001

Introduction to determinants via geometry

Norm and angle

norm[x_]  :=  (Sqrt[x  .  x])

norm[{3,  4}]

norm[{3,  4,  -2}]

angleCos[x_,  y_]  :=  x  .  (y/(norm[x] norm[y]))

angleCos[{2,  1},  {1,  2}]

angleCos[{2,  1},  {-1,  2}]

Determinant in dimension 2

Clear[a,  b,  c,  d]

A  =  { {a,  b},  {c,  d} };  A  //  MatrixForm

u  =  A[[All,  1]] ;  
    u  //  MatrixForm
v  =  A [[All,  2]];  
    v  //  MatrixForm

Needs["Graphics`Arrow`"]
Needs["Graphics`Colors`"]

Block[\({$FormatType\  = \ 
            TraditionalForm, \ $TextStyle\  = \ {FontFamily\  -> \ "Times", \ FontWeight\  -> \ "Bold", \ 
              FontSize\  -> \ 16}, 
u\  = \ {1, \ 
              0.5}, \ v\  = \ {0.4, \ 0.75}, \ delay}\),
Show[Graphics[{
Arrow[{0, 0},  u], Arrow[{0,  0},  v],
\(Text[\*""(Overscript[u,  ->]) "", \ {1, \ 0.45}]\), \(Text[\*""\!\(\(v\& -> \)\)"", \ {0.3, \ 
                        0.7}]\), Text["θ",  {0.1,  0.1}],
Red, Arrow[{0.62, 0.31},  {0.4,  0.75}], Text["∥(Overscript[v,  ->]) ∥ sin(θ)", {0.68,  0.6}]}], AspectRatio  ->  Automatic];
]

Block[\({$FormatType\  = \ 
            TraditionalForm, \ $TextStyle\  = \ {FontFamily\  -> \ "Times", \ FontWeight\  -> \ "Bold", \ 
              FontSize\  -> \ 16}, 
u\  = \ {1, \ 
              0.5}, \ v\  = \ {0.4, \ 0.75}}\),
Show[Graphics[{
Arrow[{0, 0},  u], Arrow[{0,  0},  v],
\(Text[\*""(Overscript[u,  ->]) "", \ {1, \ 0.45}]\), \(Text[\*""\!\(\(v\& -> \)\)"", \ {0.3, \ 
                        0.7}]\), Text["θ",  {0.1,  0.1}],
Red, Arrow[{0.62, 0.31},  {0.4,  0.75}], Text["∥(Overscript[v,  ->]) ∥ sin(θ)", {0.68,  0.6}],
Green, Arrow[u,  u  +  v], Arrow[v,  u  +  v]
}], AspectRatio  ->  Automatic];]

area  =  
      norm[u] norm[v] (Sqrt[1  -  (((angleCos[u,  v]))^2)])

Simplify[area]

Simplify[area, a  ∈  Reals  &&  
            b  ∈  Reals  &&  c  ∈  Reals  &&  
            d  ∈  Reals]

Det[A]

Cross-product (in 3-dimensions)

Clear[x,  y,  x1,  x2,  x3,  y1,  y2,  y3]

x  =  {x1,  x2,  x3};
y  =  {y1,  y2,  y3};

Cross[x,  y]

Type cross-product symbol as EsccrossEsc

x \[Cross] y

(x \[Cross] y)  .  x  //  Simplify
(x \[Cross] y)  .  y  //  Simplify

Determinant in dimension 3

Clear[a,  A]
A  =  Array[a,  {3,  3}] ;  A  //  MatrixForm

{u  =  A[[All, 1]],  
      v  =  A[[All,  2]],  
      w  = A[[All,  3]]};  {u  //  
        MatrixForm,  v  //  MatrixForm,  w  //  MatrixForm}

Cross[v,  w]

u  .  Cross[v,  w]  //  Expand

Det[A]

Det[A]  ==  Expand[u  .  Cross[v,  w] ]