Geometry in 3D: dot, cross, norm

First we define some utility functions;

torow = @(x)(x(:).');

tocol = @(x)(x(:));

norm (L2, Euclidean) of a vector is a scalar defined as

norm(a) == sqrt(sum(a.^2)) == sqrt(torow(a)*tocol(a)) == sqrt(dot(a,a));

dot product of two vectors is a scalar,

dot(a,b) == sum(a.*b) == torow(a)*tocol(b);

cross product of two vectors is a vector orthogonal to both, following the right hand rule:

cross(a,b) == ([a(2)*b(3)-a(3)*b(2),...

a(3)*b(1)-a(1)*b(3),...

a(1)*b(2)-a(2)*b(1)]); % each row corresponds to an axis;

Right hand rule:

(Image from Wikipedia)

Some very commonly used properties:

dot(a,b) == norm(a).*norm(b).*cos(span(a,b));
span(a,b) == acos(dot(a,b)./norm(a)./norm(b));

where span() is the angle spanned between two vectors;

norm(cross(a,b)) == norm(a).*norm(b).*sin(span(a,b));

span(a,b) == atan2(norm(cross(a,b)),dot(a,b));

This formula of span(a,b) is more robust than the acos version at near pi/2; the convention of such angle is measured by the shortest great circle path between two vectors;