# 3-Dimensional planes

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##### Intros

###### Lessons

**3-Dimensional Planes Overview:**__Equation of a Plane__- How do we get the formula for the equation?
- $a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$
- What we need for the formula

__Finding a Plane with a Parallel Plane & 1 point__- Get the Normal Vector
- Plug into the formula

__Finding the Equation of a Plane with 3 points__- Creating 2 vectors
- Using the Cross Product = Normal Vector
- Plug into the formula

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##### Examples

###### Lessons

**Finding the Equation of a Plane**

Find the equation of the plane which contains the points $(1, 3, 0)$, $(-2, 7, 6)$ and $(1, 0, 1)$.- Find the equation of the plane which contains the point $(0, -2, 1)$ and is orthogonal to the line $<1+2t, t, 0>$.
**Are the Two Planes Parallel, Orthogonal or Neither?**

Determine whether the two planes $2x+4y+6z=8$ and $x+2y+3z=1$ are parallel, orthogonal, or neither.- Determine whether the two planes $-3x+y+8z=4$ and $2x+6y=1$ are parallel, orthogonal, or neither.
**Intersection of a Plane and a Line**

Determine whether the plane $-3x+5y+z=2$ and line $r(t)=\lt2+3t, -5t, 1-t\gt$ intersect.

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###### Topic Notes

__Notes:__**Equation of a Plane**

Couple sections ago, we saw that the equation of plane can be expressed as $ax+by+cz=d$. However, this equation does not give much information. So suppose we have the following graph:

Where $\vec{r}$ and $\vec{r_0}$ are position vectors for points $P$ and $P_0$ respectively, and $\vec{n}$ is a normal vector that is orthogonal (perpendicular) to the plane.

Since $\vec{r} - \vec{r_0}$ is on the plane, then $\vec{n}$ is orthogonal to $\vec{r} - \vec{r_0}$. In other words, their dot products should give 0.

So,

$(\vec{r} - \vec{r_0}) \cdot \vec{n} = 0 \to (\lt x,y,z\gt - \lt x_0,y_0, z_0\gt) \cdot \lt a,b,c\gt = 0$

$\to \lt x-x_0 , y-y_0, z-z_0\gt \cdot \lt a,b,c\gt = 0$

$\to a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$

The key to finding the equation of a plane is finding two things:

- The normal vector (orthogonal to the plane)
- A point on the plane.

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