Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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2,819,387 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces three-dimensional coordinate systems where any point in space is represented by an ordered triple (a, b, c). It explains how the coordinate axes and planes create a framework for visualizing objects in space, including the concept of octants. Students learn to derive equations for geometric surfaces such as planes, cylinders, and spheres, and apply the 3D distance formula. Additionally, the section covers basic vector operations, demonstrating how to add vectors via the Triangle and Parallelogram Laws, which are essential for understanding displacement, force, and other vector quantities in space.

Learning Objectives

1

Explain how points are represented in three-dimensional space using ordered triples.

2

Describe the three-dimensional rectangular coordinate system, including coordinate axes, planes, and octants.

3

Derive and apply equations for surfaces such as planes, cylinders, and spheres in R³.

4

Utilize the three-dimensional distance formula and comprehend its derivation from the Pythagorean theorem.

5

Understand vector concepts and perform vector addition using both the Triangle and Parallelogram Laws.

Key Concepts

CONCEPT

DEFINITION

3D Coordinate System

A system for locating points in space using an ordered triple (a, b, c) along the x-, y-, and z-axes.

Coordinate Axes and Planes

The three mutually perpendicular lines (x-axis, y-axis, z-axis) that define space, with their corresponding planes (xy-, xz-, yz-planes) dividing space into octants.

Octant

One of the eight regions into which space is divided by the coordinate planes.

Distance Formula in 3D

A formula to calculate the distance between two points (x1, y1, z1) and (x2, y2, z2): |P1P2| = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²).

Sphere Equation

An equation of the form (x - h)² + (y - k)² + (z - l)² = r² representing all points at a fixed distance (radius r) from a center (h, k, l).

Vector

A quantity having both magnitude and direction, often represented as an arrow or a directed line segment.

Vector Addition

The process of combining two vectors using the Triangle Law or Parallelogram Law to yield a resultant vector.

Example Problems

Example 1

Suppose you start at the origin, move along the x-axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

Example 2

Sketch the points $ (1, 5, 3) $, $ (0, 2, -3) $, $ (-3, 0, 2) $, and $ (2, -2, -1) $ on a single set of coordinate axes.

Example 3

Which of the points $ A (-4, 0, -1) $, $ B (3, 1, -5) $, and $ C (2, 4, 6) $ is closest to the $ yz $-plane? Which point lies in the $ xz $-plane?

Example 4

What are the projections of the point $ (2, 3, 5) $ on the $ xy $-, $ yz $-, and $ xz $- planes? Draw a rectangular box with the origin and $ (2, 3, 5) $ as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

Example 5

What does the equation $ x = 4 $ represent in $ \mathbb{R}^2 $? What does it represent in $ \mathbb{R}^3 $? Illustrate with sketches.
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Step-by-Step Explanations

QUESTION

How do you derive the equation of a sphere with center (h, k, l) and radius r?

STEP-BY-STEP ANSWER:

Step 1: Start with the definition that a sphere is the set of all points P(x, y, z) whose distance from the center C(h, k, l) is r.
Step 2: Write the condition |PC| = r, which means √((x - h)² + (y - k)² + (z - l)²) = r.
Step 3: Square both sides to remove the square root, yielding (x - h)² + (y - k)² + (z - l)² = r².
Final Answer: The sphere is defined by the equation (x - h)² + (y - k)² + (z - l)² = r².

Finding the Equation of a Sphere

QUESTION

How do you add two vectors u and v using the Triangle Law?

STEP-BY-STEP ANSWER:

Step 1: Represent vector u and vector v with their magnitude and direction, ensuring that the terminal point of u is the initial point of v.
Step 2: Place vector v so that its tail is at the tip of u.
Step 3: The sum, u + v, is then represented by the vector from the initial point of u to the terminal point of v.
Final Answer: u + v is the resultant vector drawn from the tail of u to the tip of v.

Vector Addition Using the Triangle Law

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Common Mistakes

  • Confusing two-dimensional and three-dimensional coordinate representations; forgetting the third coordinate.
  • Misidentifying the coordinate planes or the orientation of the axes (e.g., mixing up which plane corresponds to which axes).
  • Incorrectly applying the right-hand rule in determining the positive direction for the z-axis.
  • Forgetting to square terms or properly complete the square when deriving the equation of a sphere.
  • Misinterpreting vector addition by not aligning vectors appropriately before summing their magnitudes and directions.