Book cover for Precalculus with Limits

Precalculus with Limits

Ron Larson

ISBN #9781439049099

2nd Edition

7,319 Questions

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724,371 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section extends the two-dimensional Cartesian coordinate system into three dimensions and introduces key concepts such as plotting points, calculating distances and midpoints, and formulating the equations for spheres. It further explores vectors in space, detailing how to represent them, calculate their magnitudes and perform operations like the dot product. Mastery of these topics is critical for visualizing and solving spatial problems in mathematics, physics, engineering, and other applied fields.

Learning Objectives

1

Plot and identify points in a three-dimensional coordinate system, including understanding the role of the x-, y-, and z-axes and the division into octants.

2

Apply the Distance and Midpoint Formulas in space to calculate distances between points and determine midpoints of line segments.

3

Derive and use the standard equation of a sphere, and analyze its traces with respect to the coordinate planes.

4

Understand and manipulate vectors in three-dimensional space including finding their component form, magnitude, unit vectors, and dot products.

5

Determine relationships between vectors (parallelism and orthogonality) and apply these concepts to solve geometric and real-life problems.

Key Concepts

CONCEPT

DEFINITION

Three-Dimensional Coordinate System

A spatial model formed by three mutually perpendicular axes (x, y, z) that divide space into eight regions (octants).

Coordinate Planes

The xy-, xz-, and yz-planes, each determined by two axes, which divide the three-dimensional space.

Octant

One of the eight regions in space determined by the signs of the x, y, and z coordinates; the first octant has all coordinates positive.

Distance Formula in Space

A formula to calculate the distance between two points in space: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].

Midpoint Formula in Space

A formula to determine the midpoint of a segment joining two points: ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2).

Standard Equation of a Sphere

An equation representing a sphere with center (h, k, j) and radius r: (x - h)² + (y - k)² + (z - j)² = r².

Trace

The intersection of a surface with one of the coordinate planes; for example, setting z = 0 gives the xy-trace.

Vector in Space

A directed line segment represented in component form as ⟨v₁, v₂, v₃⟩, describing both magnitude and direction.

Dot Product

An operation on two vectors u = ⟨u₁, u₂, u₃⟩ and v = ⟨v₁, v₂, v₃⟩ given by u·v = u₁v₁ + u₂v₂ + u₃v₃, used to determine the angle between them or if they are orthogonal.

Example Problems

Example 1

A _______ coordinate system can be formed by passing a $z$-axis perpendicular to both the $x$-axis and the $y$-axis at the origin.

Example 2

The three coordinate planes of a three-dimensional coordinate system are the _______, the _______, and the _______ .

Example 3

The coordinate planes of a three-dimensional coordinate system separate the coordinate system into eight _______.

Example 4

The distance between the points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ can be found using the _______ _______ in Space.

Example 5

The midpoint of the line segment joining the points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ given by the Midpoint Formula in Space is _______ .
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Step-by-Step Explanations

QUESTION

How do you compute the distance between two points, P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂), in space?

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates of P₁ and P₂.
Step 2: Compute the difference for each coordinate: Δx = x₂ - x₁, Δy = y₂ - y₁, and Δz = z₂ - z₁.
Step 3: Square each of these differences.
Step 4: Add the squared differences to obtain the sum: (Δx)² + (Δy)² + (Δz)².
Step 5: Take the square root of the sum to get the distance: d = √[(Δx)² + (Δy)² + (Δz)²].
Final Answer: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].

Distance Formula in Space

QUESTION

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

STEP-BY-STEP ANSWER:

Step 1: Recall that a sphere is the set of all points (x, y, z) that are at a distance r from the center (h, k, j).
Step 2: Use the Distance Formula in space: √[(x - h)² + (y - k)² + (z - j)²] = r.
Step 3: Square both sides of the equation to eliminate the square root.
Step 4: Write the resulting equation: (x - h)² + (y - k)² + (z - j)² = r².
Final Answer: The standard equation of the sphere is (x - h)² + (y - k)² + (z - j)² = r².

Standard Equation of a Sphere

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Common Mistakes

  • Misidentifying the orientation of the x-, y-, and z-axes, especially distinguishing between left-handed and right-handed systems.
  • Omitting the z-component in computations when extending two-dimensional formulas to three dimensions.
  • Incorrectly squaring terms or sign errors in the Distance and Midpoint Formulas in Space.
  • Confusing the equation of a sphere with that of a circle, not recognizing that a sphere represents surface points.
  • Errors in vector operations, such as miscalculating the dot product or failing to correctly determine unit vectors.