Vectors in Three Dimensions!

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distance: square root of (x2-x1)^2  +(y2-y1)^2 + (z2-z1)^2
midpoint: ( x1+x2/2 , y1+y2/2 , z1+z2/2 )
sphere: (x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2
vector equation: ( x1 y1 z1 ) = ( x0 , y0 , z0 ) + t( a , b , c )
parametric: x=x0+at  y=y0+bt  z=z0+ct
lml (magnitude): square root of x^2 + y^2 + z^2
u * v: x1 x2 + y1 y2 + z1 z2

Example 1:
Find the length and midpoint of AB.
A = ( 2 , 5 , -3 ) and B = ( 0 , 3 , 1 )
Midpoint: ( 2 + 0 / 2 , 5 + 3 / 2 , -3 + 1 / 2 ) = ( 1 , -2 , 1 )
Distance: square root of ( 0 - 2 ) ^ 2 + ( 3 - 5 ) ^ 2 + ( 1 + 3 ) ^ 2 = square root of 4 + 4 + 16 = square root of 24 = 2 square root of 6

Example 2:
Find the length and midpoint of AB.
A = ( 3 , -5 , 0 ) and B = ( -1 , 1 , 2 )
Midpoint: ( 3 + 1 / 2 , -5 + 1 / 2 , 0 + 2 / 2 ) = ( 1 , -2 , 1 )
Distance: square root of ( -1 - 3 ) ^ 2 + ( 1 + 5 ) ^ 2 + ( 2 + 0 ) ^ 2 = square root of 16 + 36 + 4 = square root of 56 = 2 square root of 14

Example 3:
Simplify the expression.
a. ( 3 , 8 , -2 ) + 2 ( 4 , -1 , 2 )
( 3 , 8 , -2 ) + ( 8 , -2 , 4 ) = ( 11 , 6 , 2 )
b. ( 1 , -8 , 6 ) * ( 5 , 2 , 1 )
5 + -16 + 6 = -5
c. magnitude of ( 3 , 5 , 1 )
square root of 3^2 + 5^2 + 1^2
= square root of 9 + 25 + 1
= square root of 35