vectortutorial solution version 2

8/21/2019 VectorTutorial Solution Version 2

1/5

EEM1016 Engineering Mathematics I: Vector Algebra

EEM1016 Engineering Mathematics ISolution - Vector Algebra

1) (a) 3 7+ = + +a b i j k , 9 1 49 59+ = + + =a b .

vector unit ara!!e! to the +a b is ( )1

3 759

+ +i j k .

(") # 0 1# 14 = + + =a b , k-j-ikji

ba $$5$4

401

31#

$$$

==

.

(c) %"serve that

# 1 3

cos 1 0 4 15

3 1 #

= = =

a b c a b c

15 15

cos 0.370614 117

= = =a b c

( )1 0cos 0.3701 6& 14 = = .(') et = d b c

$ro ro * * cos = =b c da a a d (+here$

* *

= =

d b c

dd b c

)

( ) $1

cos= a d dd

( )

( )

#

#

154 10

117

=

=

= +

da d

d

b ca b c

b c

i j k

.

(e) o!ume o- ara!!eie' having a , b an' c as e'ges

= a b c unit3 15 unit3.

(-) et # 4= + u i j k ./hen ,u a an' c !ie in a !ane o!ume 0

( )

# 4

# 1 3 0

3 1 #

= =

u a c

6 = .

#) (a) 1 1 #= + =v , ( )#

## 1 # 9 3= + + = =u

1


2/5


( ) ( ) ( )1 # 1 1 0 # 3 = + + =v u , 3 =u v

1 1 0 # #

# 1 #

= = +

i j k

v u i j k , # 1 # # #

1 1 0

= = +i j k

u v i j k

( )### # 1 3 = + + =v u ,

1 1 3cos cos3 #

= = =

u v

u v450

3com

#

= =

v

u vu

v, )$$(

#

3jiv

v.v

u.vuproj v +=

=

(") #1 1 # 6= + + =v , ( ) ( )# #

1 1 #= + =u

( ) ( ) ( )1 1 1 0 # 1 3 = + + = v u , 3 = u v

1 1 #

1 0 1

= =

i j k

v u i j k , 1 0 1

1 1 #

= = + i j k

u v i j k

( ) ( ) ( )# # #

1 1 1 3 = + + =v u ,

1 1 03cos cos 1506 #

= = =

u v

u v

3com

6

= =

v

u vu

v

, )$#$$(

6

3kjiv

v.v

u.vuproj v ++

=

= .

3) (a)(i) 1 1 1 # 3

# 1 1

i j k

u v i j k = = ,

/he area o- the ara!!e!ogram is ( ) ( )# ### 3 1 14u v = + + =

(ii) o!ume is ( )

1 1 1

# 1 1 11 # 3

u v w

= = unit

3

(")(i) 1 1 0

0 1 0

i j k

u v k = = ,

/he area o- the ara!!e!ogram is #1 1u v = =

(ii) o!ume is ( )

1 1 0

0 1 0 11 1 1

u v w = = unit3

#


3/5


4) et 0 1,#,3 , 3 7= = +r v i k . /he 'esire' !ine is

0 t= +r r v

, , 2 1, #,3 t 3,0,7the arametric euations are 1 3t, #, 2 3 7t .

5) (a) et 0 3, 3, 1 , #= = + +r v i j k . /hus, the 'esire' !ine in the -orm o-

vector euation: , , 2 3, 3, 1 t 1,1, #

arametric euation: 3 t, 3 t, 2 1 #t .

(") et 0 1,3, 1 , 0 1, 1 3,1 1 1, 4,#= = + = r v . /hus, the 'esire' !ine

in the -orm o-vector euation: , , 2 1, 3, 1 t 1, 4, #

arametric euation: 1 , 3 4 , 1 #x t y t z t= = = + .

/he arameter tin this case taes va!ues 0 1t +hen0t= , , , 1,3, 1x y z = , an' +hen 1, , , 0, 1,1t x y z = = .

8n oint "et+een 8 an' +i!! "e 'etermine' " a va!ue o- tthat !ies

"et+een 0 an' 1.

Note:

I- +e use the secon' oint as 0 0, 1,1= r , then the arametric euation

o- the !ine "ecomes , , 0, 1,1 1, 4, #x y z t= + , +ith 1 0t +hen 1, , , 1,3, 1t x y z = = , an' +hen 0, , , 0, 1,1t x y z = = ./hus +e see that "oth arametric euations 'escri"e the same !ine oining

8 an' , "ut +ith 'i--erent va!ues -or the arameter t.

(c) et 0 1,1,1=r . ince the 'esire' !ine is eren'icu!ar to the !ane

3 7 52 15 , it is ara!!e! to the vector 3,7, 5= v . ;ence,

vector euation: , , 2 1,1,1 t 3, 7, 5

arametric euation: 1 3 , 1 7 , 1 5x t y t z t= + = + =

(') ince the 'esire' !ine is eren'icu!ar to # 3= + +u i j k an'5 4 3= + +v i j k , it is ara!!e! to

1 # 3 6 1# 6

5 4 3

= = + i j k

u v i j k .

et 0 0,#,3=r . /hen the 'esire' !ine is

vector euation: , , 2 0, #, 3 t 6,1#, 6

arametric euation: 6t, # 1#t, 2 3 6t .

3


4/5


6) /he given !ines have 'irection vector 1 1, 1, 1= v an' # 3, #, 1= v ,

resective!. 8 vector eren'icu!aro+, su"stitute into the euation o- the !ane,

( ) ( ) ( )3 # 3t 7 3 7t 5 1 5t 134 0+ + + + =

t #= /hen, the oint o- intersection is -oun' " su"stituting t #= -or , an' 2.

( ) # 3 # 4= + =

( ) 3 7 # 11= + =

4


5/5


( )2 1 5 # 9= + =

/hus, the oint o- intersection is ( )4,11, 9 .

Applications1). Besu!tant -orce:

0 0 0 0

1 # 1 #

5

( cos30 cos15 ) ( sin 30 sin15 )

(3.13 0.99 ) 10 N

= + +

= +

F F F i F F j

i j

$ $

$ $

01&

316#.013.3

99.0tan

=

===

x

y

!

!

=or 'one :

5 7(3.13 0.99 ) (100 ) 10 3.13 10Nm "oule = + = F d i j i$ $ $

#).

/he 'is!acement vector r 100 i$ 50 j$ . /o -in' the +or#'one " a -orce F

'uring the 'is!acement, +e nee' to mu!ti! the 'is!acement +ith thexcomonent

o- the -orce. /hat is the +or is given " the sca!ar ro'uct,

# F.r (1#.0 i$ 70.0 j$ 96.9k$ ).(100 i$ 50 j$ )4.7>.m 4.7 ou!e

5

vectortutorial solution version 2

Documents