Chapter 10: Q. 5 (page 823)

If u and v are nonzero vectors in $ℝ^{3}$ , what is the geometric relationship between $u, v$ and $u \times v$ ?

Short Answer

Expert verified

The relation between $u, v and u \times v$ is $||u \times v|| = ||u|| ||v|| \sin θ$ ,where θ is the angle between u and v.

Step by step solution

Step 1. Given Information

If u and v are nonzero vectors in $ℝ^{3}$ , what is the geometric relationship between $u, v$ and $u \times v$ ?

Step 2. u and v are nonzero vectors in ℝ3 with the same initial point.

By Lagrange’s identity

${||u \times v||}^{2} = {||u||}^{2} {||v||}^{2} - {(u \cdot v)}^{2}$

But since $u \cdot v = ||u|| ||v|| \cos θ$ , we have

${||u \times v||}^{2} = {||u||}^{2} {||v||}^{2} - {(||u|| ||v|| \cos θ)}^{2} {||u \times v||}^{2} = {||u||}^{2} {||v||}^{2} (1 - \cos^{2} θ) {||u \times v||}^{2} = {||u||}^{2} {||v||}^{2} \sin^{2} θ$

Step 3. Then the relation between u,v and u×v is

$||u \times v|| = ||u|| ||v|| \sin θ$

where θ is the angle between u and v.

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.

Start your free trial

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

If u and v are nonzero vectors in $ℝ^{3}$ , what is the geometric relationship between $u, v$ and $u \times v$ ?

Short Answer

Step by step solution

Step 1. Given Information

Step 2. u and v are nonzero vectors in ℝ3 with the same initial point.

Step 3. Then the relation between u,v and u×v is

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

If u and v are nonzero vectors in ℝ3, what is the geometric relationship between u,v and u×v?

Short Answer

Step by step solution

Step 1. Given Information

Step 2. u and v are nonzero vectors in ℝ3 with the same initial point.

Step 3. Then the relation between u,v and u×v is

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

If u and v are nonzero vectors in $ℝ^{3}$ , what is the geometric relationship between $u, v$ and $u \times v$ ?