3D Coordinate Systems

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Introduction to 3D Coordinate Systems

3D coordinates are written in R^3, using the variables x, y, and z.

Below is a graph of y=1, a plane where all points y will equal 1.

z=y^2, a cylinder infinite along the x axis.

3D coordinates are just adding 1 more dimension to the standard 2D.

Coordinates can also be written in R^n.

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Relevant Equations

Distance Formula

sqrt((x1-x2)^2+(y1-y2)^2+z1-z2)^2)=d

Example:

Find the distance from the point (4,2-7) to the point (6,5,1).

sqrt((6-4)^2+(5-2)^2+(1-(-7))^2)=sqrt(4+9+64)=sqrt(77)

Equation of a Sphere

sqrt((x1-x0)^2+(y1-y0)^2+z1-z0)^2)=r

or

(x-x0)^2+(y-y0^2)+(z-z0)^2=r^2

Example:

Describe the surface where x^2+y^2+z^2=2x-6y+15.

=x^2-2x+1+y^2+6y+9+z^2=15+1+9 [complete the square-see notes on Math Review]

(x-1)^2+(y+3)^2+z^2=25

This is a sphere with radius 5 (sqrt(25)) centered at (1,-3,0).

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Relevant Information: Planes

Planes:

xy-plane: z=0

xz-plane: y=0

yz-plane: x=0

Example:

Graph z=3.

This is a plane where all points, z=3.

Octants

First octant: x,y,z>0

Distance to Planes

The distance to a plane can be found by using the absolute value of the coordinate.

Example:

Find the distance from (4,2,-7) to the yz-plane.

abs(4)=4.

4 is the x coordinate, the coordinate missing from the plane.