8/9/2019 Multivariable Calc Practice Exam 1

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18.02 Multivariable Calculus

Fall 2007 

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8/9/2019 Multivariable Calc Practice Exam 1

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θ

� 

18.02PracticeExam1A

Problem1. (15points) 

Aunitcube lies inthefirstoctant,withavertexattheorigin(seefigure). −−→  −−→

a)ExpressthevectorsOQ (adiagonalof thecube)andOR(joiningOtothecenterof aface) intermsof  ı̂,ˆ k. , ˆ      

  

    

     

    

O

Q

   �R

z

y

x

b)

Find

the

cosine

of 

the

angle

between

OQ

and

OR. 

Problem2. (10points) 

Themotionof apointP  isgivenbythepositionvectorR�  =3costˆ +t̂  ı+3sintˆ k. Computethevelocityandthespeedof P .

Problem3. (15points: 10,5)

a)LetA=

121

301

2−1

0

;thendet(A)=2 and A−1 = 12

1

−12

a

−22

b

5−6

;findaandb.

x

 

1

b)SolvethesystemA X =B,whereX =

 

y

andB=

−2

. z 1 

c) In the matrix A, replace the entry 2 in the upper-right corner by c. Find a value of  c forwhichtheresultingmatrixM  isnot invertible.

Forthisvalueof cthesystemM X  =0hasothersolutionsthantheobviousoneX  =0: findsuch a solution by using vector operations. (Hint: call U , V   and W  the three rows of  M , andobservethatM X =0if andonlyif X  isorthogonaltothevectorsU ,V   andW .)

Problem4. (15points)

Thetopextremityof aladderof  lengthLrestsagainstaverticalwall,while     �the

bottom

is

being

pulled

away.

Find

parametric

equations

for

the

midpoint 

P  of  the ladder, using as parameter the angle θ between the ladder and the     �P   

horizontalground.         �Problem5. (25points: 10,5,10) 

a)Findtheareaof thespacetrianglewithverticesP 0 :(2,1,0),P 1 :(1,0,1),P 2 :(2,−1,1). 

b)Findtheequationof theplanecontainingthethreepointsP 0,P 1,P 2. 

c)Findtheintersectionof thisplanewiththelineparalleltothevectorV  �  =�1,1,1andpassing throughthepointS  : (−1,0,0).

Problem

6.

(20

points:

5,

5,

10)

a)LetR�  =x(t)̂ +z(t)ˆı +y(t)̂ kbethepositionvectorof apath. Giveasimpleintrinsicformula

ford

(R�  ·R) invectornotation(notusingcoordinates).dt

b)Showthat if R�  hasconstant length,thenR�  andV  �  areperpendicular.

c)letA�  betheacceleration: stillassumingthatR�  hasconstantlength,andusingvectordifferentiation,expressthequantityR�  ·A�  intermsof thevelocityvectoronly.