3D Coordinate Systems

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.

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).

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.