Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

What is a parallelepiped? What is meant by the parallelepiped determined by the vectors u, v and w? How do you find the volume of the parallelepiped determined by u, v and w?

Short Answer

Expert verified

A parallelepiped is a three-dimensional analog of a parallelogram, in much the same way that a cube is a three-dimensional analog of a square.

The parallelepiped determined by the vectors u, v and w if they do not lie in the same plane.

Thevolumeoftheparallelepiped=|u·(v×w)|

Step by step solution

01

Step 1. Given Information 

What is a parallelepiped? What is meant by the parallelepiped determined by the vectors u, v and w? How do you find the volume of the parallelepiped determined by u, v andw?

02

Step 2. The Parallelepiped

A parallelepiped is a three-dimensional analog of a parallelogram, in much the same way that a cube is a three-dimensional analog of a square.

Specifically, a parallelepiped is a six-sided solid whose surface consists of three pairs of parallel faces, each of which is a parallelogram.

03

Step 3. The parallelepiped determined by the vectors u, v and w

Any three vectors u, v and w in 3 that do not lie in the same plane will

determine a parallelepiped.

04

Step 4. The volume of the parallelepiped

The area of a parallelogram involves a cross product. Similarly, the volume of the parallelepiped involves a triple scalar product.

Thevolumeoftheparallelepiped=u·(v×w)

if and only if u, v and w form a right-handed triple, and in any case the volume is|u·(v×w)|.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free