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3-D Cartesian Coordinates: Finding Distances and Centers of Spheres, Study notes of Mathematics

University of Pennsylvania (UPenn)Mathematics

Examples and equations for finding the distances between points and the centers and radii of spheres in three-dimensional cartesian coordinate systems. Students will learn how to calculate the distance between two points using the pythagorean theorem and find the equation of a sphere given its center and radius.

Typology: Study notes

Pre 2010

Uploaded on 03/28/2010

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Math 114 โ€“ Rimmer
13.1 3-D Cartesian Coord.
13.1 Three Dimensional Coordinate Systems (Cartesian)
Math 114 โ€“ Rimmer
13.1 3-D Cartesian Coord.
(
)
, ,
x y z
Points have coordinates:
( )
Example:
: 4,5, 6
A
(
)
: 3, 3, 1
B
โˆ’ โˆ’
(
)
Cโˆ’ โˆ’
A
B
C
pf2

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Math 114 โ€“ Rimmer 13.1 3-D Cartesian Coord. 13.1 Three Dimensional Coordinate Systems (Cartesian) Math 114 โ€“ Rimmer 13.1 3-D Cartesian Coord.

( x y z ,^ , )

Points have coordinates:

Example: A : 4,5, 6

B : 3, ( โˆ’3, โˆ’ 1 )

C : ( โˆ’2, โˆ’2,0)

A

B

C

Math 114 โ€“ Rimmer 13.1 3-D Cartesian Coord. d = PP 1 2

Distance between points: ( )

1 1 1 1 2 2 2 2 : , , : , , P x y z P x y z

2 2 2 = x 2 (^) โˆ’ x 1 (^) + y 2 (^) โˆ’ y 1 (^) + z 2 (^) โˆ’ z 1

AB = ( ) ( ( )) ( ( ))

2 2 2 4 โˆ’ 3 + 5 โˆ’ โˆ’ 3 + 6 โˆ’ โˆ’ 1 AB = 1 + 64 + 49 = 114 โ‰ˆ10.

AC = ( ( )) ( ( )) ( )

(^2 2 ) 4 โˆ’ โˆ’ 2 + 5 โˆ’ โˆ’ 2 + 6 โˆ’ 0 AC = 36 + 49 + (^36) = 121 = 11

A : 4,5,6 ( )

B : 3, ( โˆ’3,1)

C : ( โˆ’2, โˆ’2,0)

Find AB and AC. Which is larger? Math 114 โ€“ Rimmer 13.1 3-D Cartesian Coord.

(^2 2 2 ) x โˆ’ h + y โˆ’ k + z โˆ’ m = r Equation of a spheresphere :

Center: ( h k m , , ) Radius: r

Find the equation of the sphere with center at ( 0, โˆ’3, 6 )and radius 3.

2 2 2 x + y + 3 + z โˆ’ 6 = 3 2 2 2 Find the center and radius of the sphere that has the given equation: 4 x + 4 y + 4 z โˆ’ 4 x + 8 y โˆ’ 3 = 0

4 x^2 โˆ’ x + + 4 y^2^ + 2 y + + 4 z^2 = 3 + +

1 2 2 2 4 x โˆ’ 2 + 4 y + 1 + 4 z = 8 1 4 1 1 4

1 2 2 2

x โˆ’ 2 + y + 1 + z = 2 ( )

1 Center: 2 , โˆ’1,0 (^) Radius: 2