Why do we need the concept of pair of straight lines?

Question

So I started with straight lines, and I come across this concept called pair of straight lines i.e $ L_1*L_2=0$, where $L_1$ and $L_2$ are two straight lines. Now the question is what's the significance of this concept, as there is already a concept called family of lines, which says all the straight lines which are passing from the point of intersection of $L_1$ and $L_2$ can be represented as $L_1 + \lambda L_2 =0$. Is the pair of lines a subset or a particular case of family of lines? What's the use of it? Thanks in advance.

Answer 1

For two straight lines in the affine plane $\ell x+my+n=0$ and $\ell'x+m'y+n'=0$

the set of points where $(\ell x+my+n=0)(\ell'x+m'y+n'=0)=0$ is the union of the two lines. This idea can be generalized in many ways-look up algebraic sets in affine and projective space and the Zariski topology-in particlular union and intersection. In the projective plane where points are ratios $(x:y:z)$ a conic has the equation $$\begin{bmatrix}x&y&z\end{bmatrix}M\begin{bmatrix}x\\y\\z\end{bmatrix}=[0] \text { where } M=\begin{bmatrix}a&h&g\\h&b&f\\g&f&c\end{bmatrix}$$ and the conic is the union of two straight lines iff $$\begin{bmatrix}x&y&z\end{bmatrix}M\begin{bmatrix}x\\y\\z\end{bmatrix}=(\ell x+my+nz)(\ell'x+m'y+n'z) \text { for some }\ell,m,n,\ell',m',n'$$ iff $\det M=0.$ To interpret this result in the affine plane we have to make allowance for points (x:y:0) in the projective plane that do not correspond to any point(x,y) in the affine plane under the standard embedding $(x,y) \mapsto (x:y:1)$ and to the line $z=0$ of all such points in the projective plane that does not correspond to any line in the affine plane. We should also pay some attention to whether our coordinates are real numbers or complex numbers or are drawn from some other field.