Planes are *flat surfaces* in $\R^3$. Just like [[Lines|lines]], planes can be represented with a [[Lines#Vector Form|Vector Form]], [[Lines#Parametric Form|Parametric Form]], [[Lines#Normal Form|Normal Form]]. ![[../../00 Excalidraw/Planes .excalidraw.dark.svg]] # Representations ## Vector Form $\huge \alpha : (x,y,z) = P+s\vec u + t\vec v $ *Extended from [[Lines#Vector Form|Vector form of Lines]]* ## Parametric $\huge \alpha: \begin{cases} x= P_x+su_{x}+tv_{x}\\ y=P_y+su_y+tv_y\\ y=P_z+su_z+tv_z\\ \end{cases} $ *Just vector form expanded* ## Normal Form $\huge \alpha : n_{x}+n_y+n_z=c $ Where $n$ is any vector parallel to the normal of the plane. > [!tip] > If the coefficient in 0 in any $\R$, then the line passes through the original > [!example] > $ \huge \let \alpha : 3x - y - 2z = 5$ > Write $\alpha$ in vector form and in parametric form. > > $\huge\begin{align*} > \perp \alpha &= \mat{3\\-1\\-2} \\ > \end{align*}$ > > $\huge > \text{Try } x=0, y = 0 > $ > $\huge 3(0) - 0 - 2z = 5 \iff z= -\frac{5}{2}$ > $\huge \op{switch}\perp\alpha = \mat{0\\2\\-1}$ > $\huge \op{switch}\perp\alpha = \mat{1\\3\\0}$ > > *Vector Form*: > $\huge \alpha : (x,y,z) = \paren{0,0,\frac{-5}{2}} > + s\mat{1\\3\\0} + t\mat{0\\2\\-1} > $ > *Parametric Form*: > $\huge\alpha : \cases{ > x = 5 \\ > y=3s+2t \\ > z=-\frac{5}{2}-t > } > $ # Angles ## Getting the angle between a plane and a line $\huge\begin{align*} \alpha:& (x,y,z) = (3,6,-1) + t\mat{-11\\0\\2} +s\mat{1\\1\\1} \\ l:& (x,y,z) = (1,1,-1) +r \mat{3\\1\\2} \end{align*}$ Find the angle $\theta_{\angle \alpha k}$ To solve these types of problems, we can use the method described [[Lines#Angle between lines using a parallel vector and a normal vector|here]]. ![[../../00 Excalidraw/Planes _0.excalidraw.dark.svg]] Convert $\alpha$ to normal form. $\huge \mat{-1\\0\\2 }\times\mat{1\\1\\1}=\mat{2\\-3\\1} $ $\huge \alpha:2x-3y+z=-13$ $\huge\begin{align*} \vec \alpha_{\perp} &= \mat{2\\-3\\1} \\ \vec v &\parallel l\\ \vec v &= \mat{3\\1\\2} \\ \end{align*}$ $\huge\begin{align*} \theta_{\angle \alpha l} &= \arcsin \paren{\frac {\vec \alpha_{\perp}\cdot \vec v} {\norm{\vec \alpha_{\perp}}\norm{\vec v}} } \\ \\&= 20.92\degree \end{align*}$ ## Getting the angle between two planes To get the angle, you can simply find the angle between the two [[Normal Vector|Normal Vectors]] $\huge \theta_{\angle \alpha \beta } = \arccos\paren{ \frac{\vec \alpha_{\perp}\cdot \vec \beta_\perp} {\norm{\vec \alpha_\perp}\norm{\vec \beta_\perp} } } $ *Since we want the smaller, if $\theta_{\angle\alpha\beta} > 90\degree$, then take the supplement (the same as [[Lines#Getting the angle between lines|Getting the angle between lines]])*