Geometry in 3D: projection

A vector a projected onto another vector b effectively gives a vector with same orientation as b and with magnitude norm(a).*cos(span(a,b));

define

unitize = @(a)(a./norm(a));

which gives the unit vector along a;

The projection can be equivalently written in the original coordinate frame as

projection(a,b) == dot(a,b)./norm(b).*unitize(b) == dot(a,b).*b./(norm(b)^2);

Recall that

norm(b)^2 == torow(b)*tocol(b);

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

suppose a, b are 1 x 3 row vectors

We have

projection(a,b) == (a*b.').*b./(b*b.') == a*(b.'*b)./(b*b.') == a*pm;
pm == (b.'*b)./(b*b.');

pm is called the projection matrix of b;