Still Confused?

Try reviewing these fundamentals first

- Home
- Calculus 3
- Three Dimensions

Still Confused?

Try reviewing these fundamentals first

Nope, got it.

That's the last lesson

Start now and get better math marks!

Get Started NowStart now and get better math marks!

Get Started NowStart now and get better math marks!

Get Started NowStart now and get better math marks!

Get Started Now- Intro Lesson: a10:38
- Intro Lesson: b6:18
- Intro Lesson: c11:40
- Lesson: 110:16
- Lesson: 26:47
- Lesson: 35:53
- Lesson: 45:09
- Lesson: 55:02

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.

- Introduction
**3-Dimensional Planes Overview:**a)__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

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

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

- 1.
**Finding the Equation of a Plane**

Find the equation of the plane which contains the points $(1, 3, 0)$, $(-2, 7, 6)$ and $(1, 0, 1)$. - 2.Find the equation of the plane which contains the point $(0, -2, 1)$ and is orthogonal to the line $<1+2t, t, 0>$.
- 3.
**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. - 4.Determine whether the two planes $-3x+y+8z=4$ and $2x+6y=1$ are parallel, orthogonal, or neither.
- 5.
**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.