# Statement(s) > [!NOTE] Statement 1 (Intersection of two planes) > Consider the planes in $\mathbb{R}^3$, $\Pi_{1} = \{ \underline{x}\in \mathbb{R}^{3}: \underline{x} \cdot n_{1} = d_{1} \}$ and $\Pi_{2}=\{ \underline{x} \in \mathbb{R}^{3}: \underline{x} \cdot n_{2}=d_{2} \}$. If $n_{1}\times n_{2} \neq 0$, that is the normals $n_{1},n_{2}$ to the planes are not parallel, then the line of intersection is given by $L= \{ \underline{x} = c_{1} n_{1} + c_{2} n_{2} + \lambda (n_{1}\times n_{2}) : \lambda\in \mathbb{R} \}$where $c_{1}$ and $c_{2}$ are constants to be found. **Note**: if $n_{1},n_{2}$ are parallel, either $\Pi_{1},\Pi_{2}$ are equal or non-intersecting. # Proof(s) ###### Proof of statement 1: If $n_{1}\times n_{2}\neq 0$ then $\{ n_{1},n_{2}, n_{1}\times n_{2} \}$ forms a basis of $\mathbb{R}^3$. $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Reference(s)