  We introduce vector bundle techniques for finding equations of secant
varieties. A test is established that determines when a secant variety is an
irreducible component of the zero set of the equations found. We also prove an
induction theorem for varieties that are not weakly defective, that allows one
to conclude that the zero set of the equations found for s_{r-1}(X) have
s_{r-1}(X) as an irreducible component when s_r(X) is an irreducible component
of the equations found for it. The techniques are illustrated with examples of
homogeneous varieties. We give an algorithm to decompose a general ternary
quintic as the sum of seven fifth powers.
