I believe a real example will help you understand the motivation.
Consider a surface M defined by in our Euclid space . It is a cylinder. Give a point p=(1, 0, 0) on M, choose any vector v=(a,b,c), the line L defined by L(t)=p+tv = (1+at, bt, ct). When L is tangent to M, v must be a vector orthogonal to the unit vector (1, 0, 0). That is, a = 0. and p+tv=(1, bt, ct). Obviously f(L(0))=0, and
. If b=0 f(L(t)) is identically zero.