Revise: Analytic Solid Geometry
The line, the plane, and the sphere in three-dimensional coordinate space.
Every point in 3D space is (x, y, z).
P(3,4,2): 3 along x, 4 along y, 2 along z.
d = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²].
(1,2,3) to (4,6,15) → distance 13.
Parametric: x=x₀+at, y=y₀+bt, z=z₀+ct.
P₀(2,−1,3), dir (1,4,−2) → x=2+t.
a(x−x₀)+b(y−y₀)+c(z−z₀)=0.
P₀(1,2,3), n=(4,−1,2) → 4x−y+2z=8.
distance = |ax₀+by₀+cz₀−d| / √(a²+b²+c²).
Origin to x+2y+2z=3 → distance 1.
Complete the square for x, y, z separately.
x²+y²+z²−4x+6y−2z−2=0 → center(2,−3,1), r=4.
The z-axis is perpendicular to both x and y.
GPS latitude/longitude/altitude.
The 3D distance formula applies Pythagoras twice.
(2,0,4),(6,8,10) → midpoint z=7.
Use the same subtraction order for all three coordinates.
(0,5,1) to (6,1,9) → b=−4.
Symmetric form comes from solving each parametric equation for t.
x=2+t at t=3 → x=5.
The normal vector is perpendicular to every line in the plane.
d = ax₀+by₀+cz₀.
Always take the absolute value of the numerator.
(0,0,5) to 3x+4y=10 → distance 2.
The right side of the equation is the radius SQUARED.
=49 → radius 7.
Add the same constants to both sides of the equation.
x²+y²+z²+2x−4y+4=0 → radius 1.