Mathematics 261: Calculus III Guided Notes to accompany ...

Mathematics 261: Calculus III Guided Notes to accompany Thomas' Calculus

Dr. Eric Bancroft, Grove City College Spring 2021

Chapter 12 Vectors and the Geometry of Space

12.1 3-Dimensional Coordinate Systems

We move from a 2D to a 3D coordinate system by Example 1. Graph the points (2, 1, 3), (2, -1, 3), (2, 1, -3), and (-2, 1, -3)

We call 3D space

Example 2. Graph the following in R2 and R3:

(a) x = 2

(b) y = 1

(c) x2 + y2 = 4

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CHAPTER 12. VECTORS AND THE GEOMETRY OF SPACE

Recall: The distance between points P = (x1, y1) and Q = (x2, y2) in R2 was

|P Q| =

which we can prove using

Formula 1. (Distance Formula) For P = (x1, y1, z1) and Q = (x2, y2, z2) in R3 |P Q| =

Example 3. Find the distance from P = (6, -1, 1) to Q = (-2, 3, 0). Formula 2. (Equation of a Sphere) For a sphere with center (h, k, l) and radius r we have

12.1. 3-DIMENSIONAL COORDINATE SYSTEMS

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Example 4. Find an equation of the sphere which passes through the point (0, 1, -1) and is centered at (3, -1, 0).

Example 5. What is the center and radius of the sphere x2 + y2 + z2 + 4x - 2y + 9z = 2?

Example 6. For each of the following draw a picture and describe in words.

1. z < 5 2. (x - 1)2 + y2 + (z - 1)2 2 3. (x - 1)2 + y2 + (z - 1)2 > 2 4. x2 + z2 = 1

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