This lesson will extend concepts learned from Lesson 1.1, basic operations, properties, and applications of vectors, to three dimensions.
View all of the instructional videos, read the provided readings, and practice exercises/activities. These will help you master the objectives for this lesson. Pay attention to additional comments provided. This lesson is an extension of what you learned in Lesson 1.1. The only concepts that we were not introduced in Lesson 1.1 are direction angles and direction cosines.
Upon completion of lesson 1.3, you will be able to:
Identify and construct vectors in 3D
Recognize how to construct vector addition, subtraction, and scalar multiplication geometrically
Perform vector operations
Model and solve applications
View all of the following instructional videos. These will help you master the objectives for this lesson.
Vectors with 3 Components (3 Dimensions) [3:38]
Source: rootmath from YouTube
Vectors: Magnitude of a Vector 3D [10:45]
Source: ExamSolutions from YouTube
Note: More on vector notation: The presenter used two different ways to present a vector. The column notation is commonly used.More on notation of the magnitude of a vector: Sometimes, if we use "a" for a vector in a text, then we use "a" to indicate the length/magnitude/norm of the vector a. In writing, we use "a⃗ " for a vector, and "a" or "∣∣a⃗ ∣∣" or ∥∥a⃗ ∥∥ for the magnitude of a⃗ . Note that modulus also means the magnitude of a vector.
Length of a 3-Dimensional Vector [7:56]
Source: rootmath from YouTube
Note: The presenter extends the concept for finding the magnitude of a vector to beyond 3D.
Unit Vectors [6:59]
Source: KhanAcademy from YouTube
Note: In this video, the presenter introduces the formula for finding the magnitude of a n-dimensional vector.
Parallel Vectors [6:52]
Source: bullcleo1 from YouTube
Vector Addition [4:12]
Source: MsBarnett from YouTube
Note: This video is in Japanese. It is a real life experiment about vector addition and relative motion. Since most of you will have to (or have taken) physics, it is fun to watch. Read the video description first so that you know what the experiment is about. You do not need to know Japanese in order to understand what is happening.
The following required readings cover the content for this lesson. As you go through each reading, pay close attention to the content that will help you learn the objectives for this lesson.
Chapter 11. Three-dimensional Analytic Geometry and Vectors (Section 11.2 Vectors and the Dot Product in Three Dimensions)
This article provides highlights of the topics covered both in this lesson and in the next lesson. You may stop reading after Example 6 on page 5. Pay special attention to "Direction angles and direction cosines." Note the restriction: α,β,γ∈ [0,π]. Many of the examples that you see on the Internet may not follow this definition strictly. In this class, we require the condition, α,β,γ∈ [0,π]; however, there is a chance that you may need to make adjustment if necessary when you take physics class.
Jim Lambers
Jim Lambers
Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.
Review for Analytic Geometry in Three Dimensions (problems with answers)
Do #1, 6, 8, 9, 10, 11, 12, 13, 14, 15, 19
Do #5 for Lesson 1.4
Do #2, 7, 16, 17 for Lesson 1.5
Do #3, 4, 18 for Lesson 4.1
Warning: These are good problems. However, I think that the author is having trouble displaying the solutions properly on the web. When you present your work, you MUST use proper notation(s) to present a vector, otherwise, you will NOT receive full-credit.
Below are additional resources that help reinforce the content for this lesson.
This applet introduces a different way of describing a vector in 3D.
This is an interesting website. Try the Italian version and you will see that although the language has changed, the language of mathematics remains the same.
Warning: This website uses notation (a, b, c) for vectors, which is considered improper for this class. Please follow our convention, use notation (a,b,c) for vectors.
Do Homework 1.3 on MyMathLab.
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