>

restart:

 

>	with(plots):with(plottools):with(VectorCalculus):

 

>	u1:=<1,1,3>;u2:=<3,1,0>;u3:=<1,3,1>;

 

 

 

	(1)

 

>	display(arrow(<0,0,0>,u1,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,u2,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),view=[-1..3,-1..3,-1..3]);

 

>	w1:=u1/sqrt(u1.u1);

 

(2)

 

>	v2:=u2-(u2.w1)*w1;w2:=v2/sqrt(v2.v2);

 

 

	(3)

 

>

display(arrow(<0,0,0>,u1,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,u2,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow((u2.w1)*w1,v2,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow((u2.w1)*w1,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>

display(arrow(<0,0,0>,u1,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,u2,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,v2,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),arrow((u2.w1)*w1,v2,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow((u2.w1)*w1,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>	display(arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>

display(arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),implicitplot3d((w1&x w2).<x,y,z> =0,x=-1..3,y=-1..3,z=-1..3,style=surface,color=cyan),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>	v3pt1:=(u3.w1)*w1;v3pt2:=(u3.w2)*w2;

 

 

	(4)

 

>

display(arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),implicitplot3d((w1&x w2).<x,y,z> =0,x=-1..3,y=-1..3,z=-1..3,style=surface,color=cyan),arrow(<0,0,0>,v3pt1,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,v3pt2,0.1,0.4,0.2,cylindrical_arrow,color=purple),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>

display(arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),implicitplot3d((w1&x w2).<x,y,z> =0,x=-1..3,y=-1..3,z=-1..3,style=surface,color=cyan),arrow(<0,0,0>,v3pt1,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,v3pt2,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,v3pt1+v3pt2,0.1,0.4,0.2,cylindrical_arrow,color=purple),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>	v3:=u3-(v3pt1+v3pt2);

 

(5)

 

>

display(arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),implicitplot3d((w1&x w2).<x,y,z> =0,x=-1..3,y=-1..3,z=-1..3,style=surface,color=cyan),arrow(<0,0,0>,v3pt1,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,v3pt2,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,v3pt1+v3pt2,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(v3pt1+v3pt2,v3,0.1,0.4,0.2,cylindrical_arrow,color=purple),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>	w3:=v3/sqrt(v3.v3);

 

(6)

 

>

display(arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),implicitplot3d((w1&x w2).<x,y,z> =0,x=-1..3,y=-1..3,z=-1..3,style=surface,color=cyan),arrow(<0,0,0>,v3pt1,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,v3pt2,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,v3pt1+v3pt2,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(v3pt1+v3pt2,v3,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(v3pt1+v3pt2,w3,0.1,0.4,0.2,cylindrical_arrow,color=black),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>

display(arrow(<0,0,0>,u3,0.1,0.4,0.2,cylindrical_arrow,color=red),arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),implicitplot3d((w1&x w2).<x,y,z> =0,x=-1..3,y=-1..3,z=-1..3,style=surface,color=cyan),arrow(<0,0,0>,v3pt1,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,v3pt2,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,v3pt1+v3pt2,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(v3pt1+v3pt2,v3,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(v3pt1+v3pt2,w3,0.1,0.4,0.2,cylindrical_arrow,color=black),arrow(<0,0,0>,v3,0.1,0.4,0.2,cylindrical_arrow,color=purple),arrow(<0,0,0>,w3,0.1,0.4,0.2,cylindrical_arrow,color=black),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>

display(arrow(<0,0,0>,w1,0.1,0.4,0.2,cylindrical_arrow,color=blue),arrow(<0,0,0>,w2,0.1,0.4,0.2,cylindrical_arrow,color=green),implicitplot3d((w1&x w2).<x,y,z> =0,x=-1..3,y=-1..3,z=-1..3,style=surface,color=cyan),arrow(<0,0,0>,w3,0.1,0.4,0.2,cylindrical_arrow,color=black),scaling=constrained,view=[-1..3,-1..3,-1..3]);

 

>