Lesson 1.4: Dot Products

In this lesson, you will learn the definition, evaluation, interpretation, properties, and applications of the dot products.

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. Make sure that you understand the geometry of the dot product as well.

Lesson Objectives

Upon completion of lesson 1.4, you will be able to:

• Demonstrate the understanding of geometry of the dot product

• Determine dot products

• Recognize and apply properties of the dot product

• Model and solve applications involving dot products

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this lesson.

1. The Dot Product [10:33]

   1. Source: KhanAcademy from YouTube

• 1. Note: This video has physics errors.

1. Vector Dot Product and Vector Length [9:10]

   1. Source: KhanAcademy from YouTube

1. Proving Vector Dot Product Properties [10:45]

   1. Source: KhanAcademy from YouTube

• 1. Note: The presenter conveys the idea about how these properties may be proved, however, what have been presented are just sketches of proofs.

1. Dot Product – Angle Between Two Vectors [9:26]

2. Source: rootmath from YouTube

1. Determining the Angle Between Two Vectors [7:05]

   1. Source: bullcleo1 from YouTube

• • 1. Note: Before you proceed and apply the formula, you should always try to rule out the obvious. If the dot product is 0, then the two vectors are orthogonal, and therefore the angle between the two vectors is 90 degrees. In example 3, it is easy to see that the two vectors are scalar multiples of each other. Without applying the formula, you should be able to figure out the angle between the two vectors is 180 degrees.

1. Orthogonal Vectors [4:22]

   1. Source: rootmath from YouTube

• 1. Applications of the Dot Product [10:39]

  2. Source: AlRichards314 from YouTube

• • 1. Note: Instead of using the notation, a⃗ ↓b⃗ , please use notation projb⃗ a⃗ .

• 1. Orthogonal Projections – Scalar and Vector Projections [3:21]

  2. Source: patrickJMT from YouTube

Lesson Readings

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.

1. Dot Product

2. Paul's Online Math Notes

3. 13.3 The Dot Product (definition, properties, using the dot product, projections) On page 704, skip the material after Example 3. On page 705, skip Example 5. (Read what you skipped for Lesson 4.1 later.)

4. Jim Lambers

Lesson Practice Exercises/Activities

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.

1. Vectors & The Dot Product (problems with solutions)

2. Dot Product (27 practice problems with solutions)

3. Leading Lesson

4. Dot Product (5 practice problems with solutions)

Additional Resources

Below are additional resources that help reinforce the content for this lesson

1. Dot Product

   1. MathsisFun

2. Basic Operations for Vectors

Homework Assignment

Do Homework 1.4 on MyMathLab.

Show Your Work Assignment

Show Your Work assignment will be given via Assignment Tool on Laulima. Please check your email for Laulima announcement.

Lesson 1.5: Cross Products -->