In this lesson, you will learn the definition, properties, geometry, and applications of the cross 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 cross product as well. Many of the properties that you have learned about vectors hold for n-dimensional vectors, where n is an integer and n ≥ 2. But the cross product is only defined in 3-dimensional space.
Upon completion of lesson 1.5, you will be able to:
Demonstrate the understanding of geometry of the cross product
Determine cross products
Recognize and apply properties of the cross product
Model and solve applications
View all of the following instructional videos. These will help you master the objectives for this lesson.
The Cross Product [10:42]
Source: AlRichards314
Note: The presenter uses a "short cut" to obtain the cross product.
The Concept of Vector Cross Product [6:29]
Source: Daniel Kopsas from YouTube
Calculating a Cross Product Vector [7:50]
Source: Daniel Kopsas from YouTube
Producto Cruz de dos Vectores [10:53]
Source: julioprofenet from YouTube
Note: This video is in Spanish, but I guarantee you that after watching the first 3 videos you will have no problem understanding this video. I do not know Spanish but I understand every point that the presenter is trying to make. Pay special attention to the precise usage of the symbols. The way that the presenter is showing you how to calculate a cross product is the way that most (if not all) textbooks do. He does it slowly so you really can follow. Don't you agree that Mathematics is a universal language? (FYI: [a, b, c] is another way to present a vector.)
The Triple Scalar Product [4:41]
Source: bullcleo1 from YouTube
Note: This video will help you to understand the next 4 videos.
The Presenter in the following 4 videos has a strong accent, however, he does a good job at presenting the concept of scalar triple product. He shows you how the scalar triple product is defined and calculated algebraically, the properties of the scalar triple product, and the geometric meaning of the product.
Scalar Triple Product Part 1 [4:02]
Source: Wei Ching Quek from YouTube
Scalar Triple Product Part 2 [6:40]
Source: Wei Ching Quek from YouTube
Note: Instead of using the notation, a⃗ ↓b⃗ , please use notation projb⃗ a⃗ .
Scalar Triple Product Part 3 [5:21]
Source: Wei Ching Quek from YouTube
Scalar Triple Product Part 4 [4:48]
Source: Wei Ching Quek from YouTube
DSN Animation: What is Torque [0:12]
Source: DesignSquadNation from YouTube
An Application of Cross Products: Torque [7:51]
Source: bullcleo1 from YouTube
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.
Paul’s Online Math Notes
13.4: The Cross Product (the area of a parallelogram, definition of the cross product, properties of the cross product, the equivalence of the two definitions of the cross product, areas and volumes using the cross product and determinants) You may skip “The Equation of a Plane Through Three Points” and Example 4 on p 713.
Math Insight
This website has an interactive applet. It should help you to visualize some of the concepts that you have learned. Check out other links on this page (such as “The formula for the cross product”, “The scalar triple product”, …) for more applets.
Warning: The website and its related pages sometimes use (a, b, c) for a vector instead of <a, b, c> or [a, b, c]. Please do NOT use (a, b, c) to represent a vector in this course.
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.
Cross Product (16 practice problems with solutions)
Leading Lesson
Check out all the other links listed on the page, except the last link “Multivariable calculus” which includes the topics that you have not learned yet.
Cross Product Problems (6 practice problems with solutions)
Vitutor
Click on tabs 1-6 (next to the Problems tab) for solutions to the 6 problems. Explore by clicking on other tabs on top of the page.
Warning: The website and it’s related pages sometimes use (a, b, c) for a vector instead of <a, b, c> or [a, b, c]. Please do NOT use (a, b, c) to represent a vector in this course.
Cross Product Resources (10 practice problems with solutions)
17Calculus
The exercises are at the bottom of the page. The answer to the concern raised in the “Note” under the “Cross Product Definition and Notation” is “yes,” it is not really a determinant. The method given for finding the cross product is like finding the determinant of a 3x3 matrix. You will learn Linear Algebra after finish taking the sequence of Calculus courses. You may watch the two MIT lecture videos. Watch other videos linked on this page with caution, because of possible errors and/or improper notations.
Below are additional resources that help reinforce the content for this lesson.
Cross Product Discovery Applet
A Java Interactive Tutorial
Applications of the Dot Product and Cross Product (video) [7:38]
Source: AlRichards314
Visualization of Torques (Moments) (video) [5:45]
Source: udiprod
Do Homework 1.5 on MyMathLab.
Show Your Work assignment will be given via Assignment Tool on Laulima. Please check your email for Laulima announcement.
Time to turn in Show Your Work Assignment for this module.
Start taking Exam 1 as soon as possible so that you can take advantage of the opportunity of multiple attempts before the deadline. Refer to course syllabus for the policy on taking exams.