# Lecture - 7 Chapter 11 Three Dimensional Geometry

“Whenever you find yourself on the side of the majority, it is time to pause and reflect.” –Mark Twain

In this video, I am discussing equation of plane in three dimensional geometry and its derivation.

If One passing point and a normal vector is given then equation of plane in both the vector form and the Cartesian form:
$$(\vec{r}-\vec{a})\cdot\vec{n}=0$$
$$(x-x_1)a+(y-y_1)b+(z-z_1)c=0$$

If three non-collinear point are given then equation of plane in both the vector form and the Cartesian form:

$$(\vec{r}-\vec{a})\cdot[(\vec{b}-\vec{a})\times(\vec{c}-\vec{a})]=0$$
$$\left|\begin{matrix}x-x_1&y-y_1&z-z_1\\x_2-x_1&y_2-y_1&z_2-z_1\\x_3-x_1&y_3-y_1&z_3-z_1\\\end{matrix}\right|=0$$

If intercepts on the axes are given then equation of plane in the Cartesian form:
Intercept on x-axis: A
Intercept on y-axis: B
Intercept on z-axis: C
We can use identity for three passsing points to find equation of plane.
Equation of plane: $$\frac{x}{A}+\frac{y}{B}+\frac{z}{C}=1$$