3-D coordinate system

3-D coordinate system

Lessons

Notes:

3-D Coordinate System
In the 3-D Coordinate System, also denoted as R3\mathbb{R}^3, we have 3 axis (x,y,z) (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)P=(x,y,z)


Planes
The xyxy-plane corresponds to all the points at which z=0z=0.
The xzxz-plane corresponds to all the points at which y=0y=0.
The yzyz-plane corresponds to all the points at which x=0x=0.

These planes are sometimes called "coordinate planes".


Projection of a Point
When a point P=(x1,y1,z1)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=(x1,y1,z1) P=(x_1,y_1,z_1) onto the xyxy-plane will become (x1,y1,0)(x_1,y_1,0).


Distance of Two Points To find the distance between two points in R3\mathbb{R}^3, we use the formula:

D=(x2x1)2+(y2y1)2+(z2z1)2D = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

Where the two points are P=(x1,y1,z1)P=(x_1,y_1,z_1) and Q=(x2,y2,z2)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:

  1. Cylinder: (xa)2+(yb)2=r2(x-a)^2 + (y-b)^2 = r^2
    cylinder
    Points are written in the form P=(x,y,z)P=(x,y,z)
  2. Sphere: (xa)2+(yb)2+(zc)2=r2 (x-a)^2+(y-b)^2+(z-c)^2=r^2
    sphere
    Points are written in the form P=(x,y,z)P=(x,y,z)
  3. Plane: ax+by+cz=dax+by+cz=d
    plane
  • Introduction
    3-D Coordinate System Overview:
    a)
    R,R2,andR3 \mathbb{R}, \mathbb{R}^2, \mathrm{and} \;\mathbb{R}^3
    • Axis in 1D, 2D, and 3D
    • Points in 3D
    • xyxy-plane, xzxz-plane, and yzyz-plane

    b)
    Projection & Distance of points
    • Projection of a point on a plane
    • Distance between two points in 3D
    • Knowing and Applying the formula


  • 1.
    Finding Projection of Points
    Find the projection of the point (3,1,5)(-3, 1 ,5) onto the yzyz-plane.

  • 2.
    Finding the Distance of Two Points
    Find the distance between P1=(1,0,4)P_1=(1, 0, 4) and P2=(2,3,5)P_2=(-2, 3, 5).

  • 3.
    Find the distance between P1=(2,1,3) P_1=(2, -1, -3) and P2=(4,0,1)P_2=(4, 0, 1) .

  • 4.
    Graphing Equations in Different Dimensions
    Graph the equation y=4y=4 in 3D.

  • 5.
    Analyzing Equations in Different Dimensions
    Determine whether the equation (x2)2+(y1)2=25(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 R3\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 R2\mathbb{R}^2 .