EBK ELEMENTS OF ELECTROMAGNETICS
EBK ELEMENTS OF ELECTROMAGNETICS
6th Edition
ISBN: 8220101369376
Author: Sadiku
Publisher: YUZU
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Chapter 1, Problem 14P

(a)

To determine

Simplify the given expression.

(a)

Expert Solution
Check Mark

Answer to Problem 14P

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 14P

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)_.

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