GraphingCalculator 4; Window 45 21 753 1260; FontSizes 18; PaneDivider 453; BackgroundType 0; StackPanes 1; 3D.View 0.9316455529720108 -0.1175972433588964 -0.343813106374937 0.1099457889981936 0.9930607782177782 -0.0417398416518884 0.3463358012712123 0.001085934608295513 0.9381099795002995; 3D.Speed 0; Text "================================ Matemagi® by Ambjörn Naeve ©Dialectica ================================ The vector A :"; Color 5; Radius 0.1706891741071428; Expr vector(0,0,0),vector(2,0,0); Text " The vector B :"; Color 5; Radius 0.1758161272321428; Expr vector(2,0,0),vector(0,0,0)+vector(2,1,1); Text " The vector C :"; Color 5; Radius 0.1760951450892857; Expr vector(2,1-(1/30),1),vector(0,1-(1/30),1); Text "The vector D : "; Color 5; Radius 0.2001255580357143; Expr vector(0,1,1),vector(0,0,0); Text " The vector B + H is equal to the purple vector K :"; Radius 0.192138671875; Expr vector(2,0,0),vector(2,2,0); Text "The light-blue part of the roof : "; Color 5; Opacity 0.7; Expr -y+z=0,leq(0,x,2),leq(0,y,1); Text "The yellow part of the roof :"; Color 6; Opacity 0.7; Expr y+z=2,leq(0,x,2),leq(1,y,2); Text "The ”attic” floor"; Color 4; Opacity 0.7; Expr z=0,leq(0,x,2),leq(0,y)<2; Text "The vector E : "; Color 6; Radius 0.1805943080357143; Expr vector(0,1+1/30,1),vector(2,1+1/30,1); Text "The vector F : "; Color 6; Radius 0.1785365513392857; Expr vector(0,2,0),vector(0,1,1); Text " The vector G : "; Color 6; Radius 0.1850934709821428; Expr vector(2,2,0),vector(0,2,0); Text " The vector H : "; Color 6; Radius 0.1780133928571428; Expr vector(2,1,1),vector(2,2,0); Text "The vector F + D is equal to the purple vector L :"; Radius 0.1925920758928572; Expr vector(0,2,0),vector(0,0,0); Text " ANALYSIS: The light-blue vectors A, B, C, D form the light-blue half of the roof. The yellow vectors E, F, G, H form the other (yellow) half of the roof. Note that the vector E is equal to the vector A. These two vectors are only placed in different positions. Moreover, since E = -C we have E + C = 0. Note that both E and C are directed along the top of the roof. Finally we have B + H = K and F + D = L The light-blue part of the roof can now be expressed as A wedge B, and the yellow part of the roof can be expressed as E wedge H. The wedge product is distributive over addition. Hence we have: A wedge B + E wedge H = A wedge B + A wedge H = A wedge (B + H) = A wedge K. Looking at the corresponding sum of vectors, we see that the two vectors along the top of the roof cancel each other, and that each (blue-yellow) pair of slanted roof-vectors add to a (purple) horizontal vector, and that these vectors together with the other two oppositely oriented horizontal vectors give an oriented boundary of the (green) floor surface. Hence the sum of the two directed ”roof-blades” is equal to the directed ”floor-blade”. ";