Let $k$ be a field, let $F \in k[X,Y,Z]$ be a nonzero homogeneous polynomial of degree $3$, and let $C_F \subset \mathbb{P}^2_k$ be the projective plane curve cut out by $F=0$. Let $P=[X_0:Y_0:Z_0] \in C_F$ be a smooth point, and let $p=(X_0,Y_0,Z_0) \in k^3 \setminus \{0\}$ be a representative of $P$. Then the projective tangent line to $C_F$ at $P$ is the line

\begin{align*} F_X(p)X+F_Y(p)Y+F_Z(p)Z=0, \end{align*}

where $F_X,F_Y,F_Z \in k[X,Y,Z]$ denote the first partial derivatives of $F$. If $p$ is replaced by $\lambda p$ for some $\lambda \in k^\times$, then this linear equation is multiplied by $\lambda^2$, and hence defines the same projective line.