3-D coordinate system

Intro Lesson: a
13:47
Intro Lesson: b
9:53
Intro Lesson: c
7:06
Lesson: 1
1:29
Lesson: 2
2:56
Lesson: 3
2:59
Lesson: 4
4:43
Lesson: 5
4:08
Lesson: 6
3:08
Lesson: 7
5:10

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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:
3d coor system
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)
$\mathbb{R}, \mathbb{R}^2, \mathrm{and} \;\mathbb{R}^3$ $R, R^{2}, a n d R^{3}$
- 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 $\mathbb{R}^2$ .

3-D coordinate system

3-D coordinate system

Lessons

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