Distance Formula:


Find the distance between (1,5,-4) and (3,2,7)
Answer:

Midpoint Formula:

Find the midpoint between ( - 4, 5, 1 ) and ( 3, -10, 5)
Answer:
(  -4 + 3  , 5 + -10 ,  1 + 5  )  =  ( -1/2, -5/2, 6/2 ) =  (-1/2, -5/2, 3)
       2          2             2

Equation of a Sphere:

(x - h)2 + (y - k)2 + (z - j)2 = r 2   where  ( h , k , j ) is the center and r is the radius.

Find the center and radius of a sphere whose equation is  x2 + y2 + z2 + 4x - 2y + 8z + 4 = 0
Answer:
Complete the square for each variable:
(x2 +4x) + ( y2 - 2y ) +  (z2 + 8z)  =  - 4
(x2 +4x + 4) + ( y2 - 2y + 1 ) +  (z2 + 8z + 16)  =  - 4 + 4 + 1 + 16
(x + 2)2 + (y - 1)2 + (z + 64)2 = 17

Center:  (-2, 1, -64)   Radius:

Traces:

To find an xz trace, let y = 0.
To find an xy trace, let z = 0.
To find a yz trace, let z = 0.

Find the xy-trace, the yz-trace, and xz-trace for the sphere whose equations is (x - 2)2 + (y + 3)2 + (z - 4)2 = 25

Answer:
Let y = 0
(x - 2)2 + (0 - 1)2 + (z - 4)2 = 25
(x - 2)2 + 1 + (z - 4)2 = 25
(x - 2)2 + (z - 4)2 = 24
The xz-trace is a circle whose center is (2,4) and whose radius is 

Let z = 0
(x - 2)2 + (y - 1)2 + (0 - 4)2 = 25
(x - 2)2 + (y - 1)2 + 16 = 25
(x - 2)2 + (y - 1)2  = 9
The xy-trace is a circle whose center is (2,1) and whose radius is 3.

Let x = 0
(0 - 2)2 + (y - 1)2 + (z - 4)2 = 25
4 + (y - 1)2 + (z - 4)2 = 25
(y - 1)2 + (z - 4)2 = 21
The yz-trace is a circle whose center is (1,4) and whose radius is 
 
 
 


Problems(Questions are there, but no answers are posted)


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