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UNIVERSIDADE FEDERAL DO CEARCENTRO DE CINCIAS E TECNOLOGIA

DEPARTAMENTO DE ENGENHARIA DE TELEINFORMTICA1 LISTA DE EXERCCIOS: ELETROMAGNETISMO

PROF. JOO BATISTA

1. Find the unit vector along the line joining point (2, 4, 4) to point ( - 3 , 2, 2).

2. Let A = 2ax + 5ay, - 3az , B = 3ax - 4ay, and C = ax + ay+ az. (a) DetermineA + 2B. (b) Calculate |A - 5C|. (c) For what values ofkis |kB| = 2? (d) Find(A X B)/(A B).

3. If=2 + 3

=

=3 5 7

x y z

y z

x y z

+ +

A a a a

B a a

C a a a

Determine:

(a) A - 2B + C(b) C - 4(A + B)

(c )2 -3A B

C

(d) A C - |B|2

(e)1 1 1

+2 3 4

B A C

4. If the position vectors of points T and S are 3 2x y z +a a a and 4 6 2x y x+ +a a a ,respectively, find: (a) the coordinates ofTand S, (b) the distance vector from Tto S, (c) the

distance between Tand S.

5. If=5 +3 +2

B= +4 +6

=8 2

x y z

x y z

x y

+

A a a a

a a a

C a a

find the values ofand such that + + A B C is parallel to the y-axis.

6. Given vectors= + +4

=3 + +6

=5 2

x y z

x y z

x y z

+

A a a a

B a a a

C a a a

determine , , and such that the vectors are mutually orthogonal.

7. (a) Show that

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( ) ( ) ( )2 2 2+ =A B A B ABg

(b) Show that

, ,y z x yz xx y zx y z x y z x y z

= = =

a a a aa aa a a

a a a a a a a a ag g g

8. Given that=2

=4 +3 +2

= 2 2

x y z

x y z

x y zC

+ +

P a a a

Q a a a

a a a

find: (a) |P + Q - R|, (b) P Q Rg , (c) Q P Rg , (d) ( ) ( ) P Q Q R g , (e) ( ) ( ) P Q Q R ,(f) cos PR , (g) sin PQ .

9) Given vectors T = 2ax 6ay + 3az, and 2x y z+ +a a a , find: (a) the scalar projection ofT on S, (b) the vector projection of S on T, (c) the smaller angle between T and S.

10. If = 6 5x y z + +A a a a and 2 3x y z= + +B a a a , find: (a) the scalar projections of

A on B, (b) the vector projection of B on A, (c) the unit vector perpendicular to the planecontaining A and B.

11. Express the following points in Cartesian coordinates:

) (1,60,2)

) (2,90 , 4)

) (4, 2, 6)

a P

b Q

c T

12. Express the following points in cylindrical and spherical coordinates:) (1, 4, 3)

) (3,0,5)

) ( 2,6,0)

a P

b Q

c R

13. (a) IfV xz xy yz = + , express Vin cylindrical coordinates.(b) If 2 2 22 3U x y z = + + , express Uin spherical coordinates.

14. Transform the following vectors to cylindrical and spherical coordinates:

2 2 2 2

( ) ( )

( ) ( ) ( )

y

x y z

a x z

b y x xyz x z

= +

= + +

D a

E a a a

15. Convert the following vectors to cylindrical and spherical systems:

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2 2 2

2 2

2 2 2 2 2 2 2 2 2

4( )

( ) ( )

x y z

yx z

x ya

x y z

yx zb x y

x y z x y z x y z

+ +=

+ +

= + + +

+ + + + + +

a a aF

aa aG

16) Express the following vectors in Cartesian coordinates:2( ) ( 1) cos

( ) 2 sin cos cos cos sinr

a z z

b r r r

= +

= +

A a a

B a a a

17. Convert the following vectors to Cartesian coordinates:

2 2

( ) sin cos 2

sin cos( )

z

r

a z z

b Dr r

= +

= +

C a a a

a a

18. Prove the following:

( ) cos

sin

sin

cos

( ) sin cos

cos cos

sin sincos sin

cos

sin

x

x

y

y

x r

x

y r

y

z r

z

a

b

=

=

=

=

=

=

==

=

=

a a

a a

a a

a a

a a

a a

a aa a

a a

a a

g

g

g

g

g

g

g

g

g

g

19. Calculate the distance between the following pairs of points:( )(2,1,5) (6, 1,2)

( )(3, 2, 1) (5,3 2,5)

)(10, 4,3 4) (5, 6 ,7 4)

a and

b and

c and

20. Given vectors 2 4 10 and 5 3x y z z = + + = + A a a a B a a a , find( ) at (0,2, 5)

( )The angle between and at P

(c) The scalar component of along at P

a P

b

+ A B

A B

A B

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21. Using the differential length dl, find the length of each of the following curves:

( ) 3, 4 2, constante

( ) 1, 30 ,0 60

( ) 4,30 90 , constante

a z

b r

c r

= < < == =