Convert a line through two points without dividing by zero

$\mathbf m=(5,-4,6)-(3,-4,7)=(2,0,-1)$. A vector equation is $\mathbf r=(3,-4,7)+\lambda(2,0,-1)$, $\lambda\in\mathbb R$.

Equating components gives $(x,y,z)=(3+2\lambda,-4,7-\lambda)$, so $x=3+2\lambda$, y=−4 and $z=7-\lambda$.

From the x-coordinate, $\lambda=(x-3)/2$; from z, $\lambda=7-z$. Thus $(x-3)/2=7-z$, together with y=−4. The zero direction component fixes y: it must not appear as a division by 0. More generally, if m₃=0 while m₁,m₂ are nonzero, write $(x-a_1)/m_1=(y-a_2)/m_2$ and z=a₃.