Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 1, Problem 15P

(a)

To determine

Simplify the given expression.

(a)

Expert Solution
Check Mark

Answer to Problem 15P

The simplified form of the expression is (BA)A(AA)B_.

Explanation of Solution

Calculation:

Consider the following two vectors.

A=(Ax,Ay,Az)        (1)

B=(Bx,By,Bz)        (2)

Calculate the cross product A×B using Equations (1) and (2).

A×B=[axayazAxAyAzBxByBz]

A×B=(AyBzAzBy)ax(AxBzAzBx)ay+(AxByAyBx)az        (3)

Calculate the cross product A×(A×B) using Equations (1) and (3).

A×(A×B)=[axayazAxAyAz(AyBzAzBy)(AxBzAzBx)(AxByAyBx)]={[(AxByAyBx)Ay+(AxBzAzBx)Az]ax[(AxByAyBx)Ax(AyBzAzBy)Az]ay+[(AxBzAzBx)Ax(AyBzAzBy)Ay]az}={[(AxAyByAyAyBx)+(AxAzBzAzAzBx)]ax[(AxAxByAxAyBx)(AyAzBzAzAzBy)]ay+[(AxAxBzAxAzBx)(AyAyBzAyAzBy)]az}={[AxAxBx+AxAyBy+AxBzAz]ax+[AyAzBz+AyAyBy+AzAzBz]ay+[AxAzBx+AyAzBy+AzAzBz]az[AxAxBx+AyAyBx+AzAzBx]ax[AzAzBz+AyAyBy+AxAxBy]ay[AxAxBz+AyAyBz+AzAzBz]az}

Simplify the equation as follows:

A×(A×B)=(AxBx+AyBy+AzBz)(Ax,Ay,Az)(AxAx+AyAy+AzAz)(Bx,By,Bz)=(AB)A(AA)B=(BA)A(AA)B{AB=BA}

Conclusion:

Thus, the simplified form of the expression is (BA)A(AA)B_.

(b)

To determine

Simplify the given expression.

(b)

Expert Solution
Check Mark

Answer to Problem 15P

The simplified form of the expression is A2(A×B)_.

Explanation of Solution

Calculation:

Consider that A×B=C.

Then, the expression for A×[A×(A×B)] is,

A×[A×(A×B)]=A×[A×C]        (4)

Consider the expression for vector triple product.

A×(A×C)=A(AC)C(AA)        (5)

Substitute Equation (5) in (4).

A×[A×(A×B)]=A(AC)C(AA)=A(A(A×B))(A×B)(AA){C=A×B}

A×[A×(A×B)]=A(A(A×B))(A×B)(AA)        (6)

Calculate the cross product A(A×B) using Equations (1) and (3).

A(A×B)=(Ax,Ay,Az)((AyBzAzBy),(AxBzAzBx),(AxByAyBx))=Ax(AyBzAzBy)(AxBzAzBx)Ay+(AxByAyBx)Az=(AxAyBzAxAzBy)(AxAyBzAyAzBx)+(AxAzByAyAzBx)=0

Substitute 0 for A(A×B) in Equation (6).

A×[A×(A×B)]=A(0)(A×B)(AA)=(A×B)(AA)=A2(A×B)

Conclusion:

Thus, the simplified form of the expression is A2(A×B)_.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Knowledge Booster
Background pattern image
Mechanical Engineering
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Text book image
Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press
Text book image
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON
Text book image
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Text book image
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Text book image
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Text book image
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY
Thermodynamics: Maxwell relations proofs 1 (from ; Author: lseinjr1;https://www.youtube.com/watch?v=MNusZ2C3VFw;License: Standard Youtube License