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# 3-D coordinate system

- Intro Lesson: a13:47
- Intro Lesson: b9:53
- Intro Lesson: c7:06
- Lesson: 11:29
- Lesson: 22:56
- Lesson: 32:59
- Lesson: 44:43
- Lesson: 54:08
- Lesson: 63:08
- Lesson: 75:10

### 3-D coordinate system

#### Lessons

__Notes:____3-D Coordinate System__

In the 3-D Coordinate System, also denoted as $\mathbb{R}^3$, we have 3 axis $(x,y,z)$. We draw the axis' in the graph like this:

Points are written in the form $P=(x,y,z)$

__Planes__

The $xy$-plane corresponds to all the points at which $z=0$.

The $xz$-plane corresponds to all the points at which $y=0$.

The $yz$-plane corresponds to all the points at which $x=0$.

These planes are sometimes called "coordinate planes".

__Projection of a Point__

When a point $P=(x_1,y_1,z_1)$ gets projected onto a plane, then the point will be on the plane.

For example, the projection of point $P=(x_1,y_1,z_1)$ onto the $xy$-plane will become $(x_1,y_1,0)$.

** Distance of Two Points**
To find the distance between two points in $\mathbb{R}^3$, we use the formula:

$D = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Where the two points are $P=(x_1,y_1,z_1)$ and $Q=(x_2,y_2,z_2)$.__Other General Equations in 3D__

Here are the general types of equations you might see in the 3-D Coordinate system:

- Cylinder: $(x-a)^2 + (y-b)^2 = r^2$

Points are written in the form $P=(x,y,z)$ - Sphere: $(x-a)^2+(y-b)^2+(z-c)^2=r^2$

Points are written in the form $P=(x,y,z)$ - Plane: $ax+by+cz=d$

- Introduction
**3-D Coordinate System Overview:**a)$\mathbb{R}, \mathbb{R}^2, \mathrm{and} \;\mathbb{R}^3$

- Axis in 1D, 2D, and 3D
- Points in 3D
- $xy$-plane, $xz$-plane, and $yz$-plane

b)__Projection & Distance of points__

- Projection of a point on a plane
- Distance between two points in 3D
- Knowing and Applying the formula

c)__Graphing & Analyzing Equations__

- Graphing the equation $x=3$ in $\mathbb{R}, \mathbb{R}^2, \mathrm{and}\; \mathbb{R}^3$
- Can this equation be graphed in this dimension?
- Other Equations we might see in the course

- 1.
**Finding Projection of Points**

Find the projection of the point $(-3, 1 ,5)$ onto the $yz$-plane. - 2.
**Finding the Distance of Two Points**

Find the distance between $P_1=(1, 0, 4)$ and $P_2=(-2, 3, 5)$. - 3.Find the distance between $P_1=(2, -1, -3)$ and $P_2=(4, 0, 1)$.
- 4.
**Graphing Equations in Different Dimensions**

Graph the equation $y=4$ in 3D. - 5.
**Analyzing Equations in Different Dimensions**

Determine whether the equation $(x-2)^2+(y-1)^2=25$ can be graphed in 1D, 2D and 3D. - 6.
**Express a 3D Shape as an Equation**

Suppose a sphere is centred at (2,5,3) and the radius of the sphere is 4. Express the sphere as an equation in $\mathbb{R}^3$. - 7.Suppose a cylinder is centred at the origin, and the radius of the cylinder is 3 cm. In addition, the cylinder is aligned to the y-axis. Express the cylinder as an equation in $\mathbb{R}^2$.