Existence: A Unique Pt Through Two Distinct Lines in the R. Proj. Plane

The Existence of a Unique Point Through Two Distinct Lines in the Real Projective Plane

Recall from The Existence of a Unique Line Through Two Distinct Points in the Real Projective Plane page that if $\mathbf{p} = [p_1, p_2, p_3]$ and $\mathbf{q} = [q_1, q_2, q_3]$ are two distinct points in the real projective plane, then there exists a unique line $<a_1, a_2, a_3>$ that passes through these points, namely, there exists unique real numbers $a_1, a_2, a_3 \in \mathbb{R}$, not all zero, such that:

(1)

\begin{align} \quad a_1p_1 + a_2p_2 + a_3p_3 = 0 \\ \quad a_1q_1 + a_2q_2 + a_3q_3 = 0 \end{align}

We will now prove an analogous proposition which will tell us that if $<a_1, a_2, a_3>$ and $<b_1, b_2, b_3>$ aretwo distinct lines in the real projective plane, then there exists a unique point $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}^2 (\mathbb{R}^2)$ through these lines.

Proposition 1: Let $<a_1, a_2, a_3>$ and $<b_1, b_2, b_3>$ be two distinct lines in the real projective plane. Then there exists a unique point $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}^2 (\mathbb{R})$ through these two lines.

The proof of this proposition is much like the proof in the proposition on the previous page showing the existence of a unique line through two distinct points in the projective plane.

• Proof: Let $<a_1, a_2, a_3>$ and $<b_1, b_2, b_3>$ be two lines in the real projective plane. For $x_1, x_2, x_3 \in \mathbb{R}$, not all zero, consider the system of equations:

(2)

\begin{align} \quad a_1x_1 + a_2x_2 + a_3x_3 = 0 \\ \quad b_1x_1 + b_2x_2 + b_3x_3 = 0 \end{align}

• We want to show that there exists unique numbers $x_1, x_2, x_3 \in \mathbb{R}$ (not all zero) so that the lines $<a_1, a_2, a_3>$ and $<b_1, b_2, b_3>$ both contain $\mathbf{x} = [x_1, x_2, x_3]$. Consider the system above written in terms of matrices:

(3)

\begin{align} \quad \begin{bmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix} \end{align}

• Since $<a_1, a_2, a_3>$ and $<b_1, b_2, b_3>$ are distinct lines, we have that the rows of $\begin{bmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{bmatrix}$ are linearly independent, so for the system above, there must exist a unique nontrivial solution $\mathbf{x}$ to this system, i.e., there exists a unique point $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}(\mathbb{R}^2)$ where $x_1, x_2, x_3$ are not all zero that is on the lines $<a_1, a_2, a_3>$ and $<b_1, b_2, b_3>$.