%% Examples from class Friday, Jan. 31

%% area of parallelogram in 2D
% For the vectors a = (-1,3) and b = (5,1) find the area of the parallelogram using
% (i) the general formula, (ii) the determinant. 

a = [-1,3]; b = [5,1];

Asq = dot(a,a)*dot(b,b)-dot(a,b)^2      % A^2
area1 = sqrt(Asq)

determinant = det([a;b])                % note that vectors a,b have negative orientation                     
area2 = abs(determinant)

or = [0,0];
gray = [.9 .9 .9];                      % RGB color gray
fillpts([or;a;a+b;b],gray); hold on
arrow(or,a,'r'); arrow(or,b,'b')
texts(a,'a'); texts(b,'b')
hold off; axis equal; grid on


%% area of parallelogram in 3D
% For the vectors a = (1,2,3) and b = (-3,-1,0) find the area of the parallelogram using
% (i) the general formula, (ii) the cross product

a = [1,2,3]; b = [-3,-1,0];

Asq = dot(a,a)*dot(b,b)-dot(a,b)^2      % A^2
area1 = sqrt(Asq)

x = cross(a,b)                          % cross product
area2 = norm(x)
          
or = [0,0,0];
fillpts([or;a;a+b;b],gray); hold on
arrow3(or,a,'r'); arrow3(or,b,'b'); arrow3(or,x,'g')
texts(a,'a'); texts(b,'b'); texts(x,'x')
hold off; nice3d

