> [!NOTE] Cartesian equation of line in $\mathbb{R}^{2}$ > $ax+by=c, \quad (a,b)\neq(0,0).$ ^393c09 > [!NOTE] Cartesian equation of plane in $\mathbb{R}^{3}$ > $ax+by+cz=d, \quad (a,b,c)\neq(0,0,0).$ > [!NOTE] Vector calculus equation of plane $\Pi$ in $\mathbb{R}^{3}$ > $\underline{\hat{n}} \cdot (x,y,z)^{T} = d$ for some $d\in \mathbb{R}$. $d$ is the height of the $\Pi$ above the origin. >> The normal vector $\underline{\hat{n}}$ determines the slope of the plane, while the value $d$ determines its distance from the origin so Choosing a different value of $d$ also defines a plane, but a different one that is parallel to $\Pi$. > [!NOTE] Parametrisation of line in $\mathbb{R}^{3}$ > The line $L$ through $P$ that is parallel to $\underline{w}$ is described as the set $L=\{ P+\lambda \underline{w} \mid\lambda\in\mathbb{R} \}$. > [!NOTE] Cartesian equation of line in $\mathbb{R}^{3}$ > We need more than equation of the form $ax+by+cz=d, \quad (a,b,c)\neq(0,0,0)$ by theory of dimensions. > >Note that a line is an intersection of two planes.