Q. 66 Use the definition of the cross ... [FREE SOLUTION]

Chapter 10: Q. 66 (page 825)

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u 脳 v = 鈭� v 脳 u$ . (This is
Theorem 10.27.)

Short Answer

Expert verified

Hence, we prove that $u 脳 v = 鈭� (v 脳 u)$ .

Step by step solution

Step 1. Given Information

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u 脳 v = 鈭� v 脳 u$ . (This is Theorem 10.27.)

Step 2. Theorem 10.27

Theorem states that "For any vectors u and v in ${鈩�}^{3}$ $u 脳 v = 鈭� (v 脳 u)$ ."

Step 3. We have to prove u×v=−(v×u)

Firstly finding the value of $u 脳 v$

role="math" localid="1649916954076"

u 脳 v = (u_{1}, u_{2}, u_{3}) 脳 (v_{1}, v_{2}, v_{3}) u 脳 v = [\begin{matrix} i & j & k \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \end{matrix}] u 脳 v = i |\begin{matrix} u_{2} & u_{3} \\ v_{2} & v_{3} \end{matrix}| - j |\begin{matrix} u_{1} & u_{3} \\ v_{1} & v_{3} \end{matrix}| + k |\begin{matrix} u_{1} & u_{2} \\ v_{1} & v_{2} \end{matrix}| u 脳 v = i (u_{2} v_{3} 鈭� u_{3} v_{2}) - j (u_{1} v_{3} - u_{3} v_{1}) + k (u_{1} v_{2} 鈭� u_{2} v_{1}) u 脳 v = (u_{2} v_{3} 鈭� u_{3} v_{2}, 鈭� u_{1} v_{3} + u_{3} v_{1}, u_{1} v_{2} 鈭� u_{2} v_{1})

Step 4. Now finding the value of v×u

$v 脳 u = (u_{1}, u_{2}, u_{3}) 脳 (v_{1}, v_{2}, v_{3}) v 脳 u = [\begin{matrix} i & j & k \\ v_{1} & v_{2} & v_{3} \\ u_{1} & u_{2} & u_{3} \end{matrix}] v 脳 u = i |\begin{matrix} v_{2} & v_{3} \\ u_{2} & u_{3} \end{matrix}| - j |\begin{matrix} v_{1} & v_{3} \\ u_{1} & u_{3} \end{matrix}| + k |\begin{matrix} v_{1} & v_{2} \\ u_{1} & u_{2} \end{matrix}| v 脳 u = i (v_{2} u_{3} 鈭� v_{3} u_{2}) - j (v_{1} u_{3} - v_{3} u_{1}) + k (v_{1} u_{2} 鈭� v_{2} u_{1}) v 脳 u = (v_{2} u_{3} 鈭� v_{3} u_{2}, 鈭� v_{1} u_{3} + v_{3} u_{1}, v_{1} u_{2} 鈭� v_{2} u_{1}) v 脳 u = 鈭� (u_{3} v_{2} 鈭� u_{2} v_{3}, 鈭� u_{3} v_{1} + u_{1} v_{3}, u_{2} v_{1} 鈭� u_{1} v_{2})$

Hence it shows that $u 脳 v = 鈭� (v 脳 u)$

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 91影视!

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u 脳 v = 鈭� v 脳 u$ . (This is
Theorem 10.27.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Theorem 10.27

Step 3. We have to prove u×v=−(v×u)

Step 4. Now finding the value of v×u

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

91影视

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that u脳v=鈭�v脳u. (This isTheorem 10.27.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Theorem 10.27

Step 3. We have to prove u×v=−(v×u)

Step 4. Now finding the value of v×u

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u 脳 v = 鈭� v 脳 u$ . (This is
Theorem 10.27.)