diff --git a/docs/mcv4u7.md b/docs/mcv4u7.md index 9fdc0f1..65e009a 100644 --- a/docs/mcv4u7.md +++ b/docs/mcv4u7.md @@ -556,6 +556,32 @@ $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$ $$c^2=a^2+b^2-2ab\cos C$$ +### Vector planes + +A **plane** ($\pi$) is a flat surface extending infinitely in all directions and can be represented by a parallelogram. + +So long as any of the following are known, their plane can be constructed: + + - Two intersecting or parallel lines + - A line and a point not on the line + - 3 non-colinear points + +In each scenario, an initial point $r_0$ and two direction vectors $\vec{u},\vec{v}$ can be derived to form the equation for a plane: +$$\vec r = \vec r_0 + s\vec u + t\vec v,s,t\in\mathbb R$$ + +This can be expanded to form the parametric form of the equation: +$$ +x=x_0+su_1+tv_1 \\ +y=y_0+xu_2+tv_2 \\ +z=z_0+su_3+tv_3,s,t\in\mathbb R +$$ + +Where $A,B.C.D$ are all integers, the **scalar** or Cartesian equation of a plane in three dimensions can be expressed as follows: +$$Ax+By+Cz+D=0$$ + +!!! info + $[A,B,C]$ is the **normal direction vector** of a plane. + ## Resources - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)