12.1 Three Dimensional Coordinate Systems, Summaries of Calculus

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Math 232

Calculus III

Brian Veitch • Fall 2015 • Northern Illinois University

12.1 Three Dimensional Coordinate Systems

Definition 1: Point

A point P (a, b, c) is to be understand as

a = x − coordinate

b = y − coordinate

c = z − coordinate

If x = 0 ⇒ yz plane. If y = 0 ⇒ xz plane. If z = 0 ⇒ xy plane.

• 2. x =
• 3. y = −

Formula 1: Distance and Midpoint Between Points

Given two points P (x 1 , y 1 , z 1 ) and Q(x 2 , y 2 , z 2 ) the distance between the two points is

Distance: |P Q| =

(x 1 − x 2 )^2 + (y 1 − y 2 )^2 + (z 1 − z 2 )^2

Midpoint: M id(P Q) =

x 1 + x 2 2

y 1 + y 2 2

z 1 + z 2 2

Example 3

Find the midpoint and distance between the points P (2, − 1 , 7) and Q(1, − 3 , 5)

|P Q| =

(1 − 2)^2 + (−3 + 1)^2 + (5 − 7)^2 =

M id(P Q) =

2. −x + y = 5. There is no restriction on z meaning z can be anything. This turns the

line from the previous example into a plane.

3. x^2 + y^2 = 9, z = 2. In the xy plane this is a circle with radius 3. Shift the circle up to

z = 2.

4. x^2 + 9^2 = 9. Since there is no restriction on z the circle can extend up and down along

the z axis to form a cylinder.

Definition 3: Equation of a Sphere