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Exam 1 L. Ballou Name____________________________ Math 231 Calculus III September 18, 2015 1. (4 points) Find the equation of the plane containing the points � �2,4,3 , � ��3, 5,0

and � ��4,1,6 . 2. (4 points) Find the angle of intersections of the planes � � 2 3 4x y z and

� � � 2 4 6x y z . 3. (8 points) For � �3 4u j k and � � �4 5v i j k , determine the projuv . 4. (8 points) For 3,4,0v and 2,3,1u , let find a vector orthogonal to both u

and v with length 3.

Solution

P#=< 1,

. q, -3>P#P&|Is¥e±÷/= . iattsj - iak

P#< -6 . -3,3 > Plane : -121×-2 )t5ly- 4) -1913-37=0OR 12×-59+192=61

OR -36×+159-572=-183

T,=< 1,2 ,z > TV, 'Tz=< 1.2.37 's -2,1 , -47=(1×-2)+411 )+3t⇒= . )z

NF< -2,1 , -4 > So Cos D= DINI =-12118,11--F4ta=| IFMHIIFVZHTHFI

" NTl=FtE=r O - cos' YEE)

= *,

pr%t=comPF¥u, ,=" tncosotuta ,=tETY÷#,Jirproffr

#i=at6=5T=< Q

, -3,4 > =f±jt< O, -3,47

T=< -4,1 ,5 > Tl#=0tHt -34+4151=17 =¥s< D, -3,4 >

Now TtxT=h -4,3 , -1 >

or< 4, -3,17 is Orthogonal toN×T⇐g&g¥o|=f4 , -13,

- ' > both -u4T

To get length } :

11 'uxTH=TtH=2T÷ as -4,3 ,-17

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5. (4 points) If �3, ,2u x and �,1, 2w x , find x such that u and w are orthogonal.

6. (6 points) Find parametric equations for the line through the points � ��1, 2,3P

and � ��4, 1,2Q . 7. (8 points) Find the area of the parallelogram with vertices at the points � �1,1,1R ,

� �2,1,3Q and � ��3, 1,1P . 8. (8 points) Find an equation of the line tangent to the curve � � 2 3, ,r t t t t at 1.t

To be orthogonal ti .D=o,

3×-4-4=0 # 24=4 X=2

Need a direction uectorforahne : P#T=< 3,1 ,- / >

vector form : Fltth I, -2,3 >+t< 3,1 , -1 > TER

= < 1+3t,

-2ft , 3 - t >

Parametric form : X=H3t y= -2++3=3 - t tee

POT - +1,2 , 2 >

91.117115' 't FR= { -2,2, 0 >

8 '

Rpaidth poJ×ppT=( 4. -4,2 >

but "A= HPTXFQH -

- ¥+4A= bihtltnllhllsinottuxvll =r}6= 6

Fu )=< 41,1 >F '=< 1,2T . 3+2 >

F '

( l ) = ( 1, 2,3 >

Ilt)=< 1, 1,1 >+t< 1,213 >QR< 1,1 , httgnkhz, } >

=< Itt .

l+2t ,1+3+7

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9. (8 points) If the velocity of a particle is � � � �2 2, , 1tv t t e t , find the position

vector � �r t if � � �0 1,4, 1r .

10. (8 points) Find the length of the curve � � � ��§ ·� �¨ ¸© ¹

3/22 2 122 3

tttr i j over d d0 6t

11. (4 points) Describe/sketch the quadric surface given by �2 2

2

9 9y zx .

12. (8 points) Show that the planes given by � � 2 4 3 5x y z and � � � 4 8 6 1x y z are

parallel, and find the distance between the planes.

at )=JtHdt=(t÷tG,

- tsinttcz ,§tt+D" tcs >

Tlothotci , -5+1 , }+g>=< 1,4, -1 >

C ,=l - E+G=4 ÷tC}= 't lnfltfftftlitsettasisttth "is⇒cz=Qz Cs= - I

tH¥( Isl , tslzttd " >

11711=+44's 's 't,

t.gl#+yy..z,/=tt+,t2t+'=Ftt=Lt, 12++11427

l=fbLt+Ddt=tEttl!=3¥k 6--18+6=24

Cone - opens along

taxi1,0,- D Rtha ' 'k)

D,=< 2 , -4,3>

RJR.=< -2,0 ,Yz>Des -4,46 >

SONI - IB ,thus theplanesare parallel .

othrbesIN

.FR#ltto-3kl_=±

y y ' " plane '

MATH kg 2529

-

nrthisis .

- -+

length betweenEC,nµamz planets .

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13. (8 points) Show that the lines determined by � �� �

� �

� � �1

2

7,3 4 ,2 6

6 ,3 8 ,9 5

r t t t

r t t t t

Intersect, then find an equation of the plane containing the two lines.

14. (8 points) Find the unit tangent vector, ,T the unit normal vector N for the

vector-valued function � � � � � � � �� �= 3 sin sin 2cost t t tr i j k ..

T, -0

,-4 ,

6 >

# < -1,8 ,-5 >

x : 7=6 -

t⇒±- ' #T×#fd

,Iy&g|y :3-45-35584⇒ -45=-8 s=2

z : 2+65=9 - 5t check 2+64=9 -5T ' )- 14=14 =L - 28

, -6 , -47=-244, 3,27

Inkom itheyntersedyplane ,

T ,H=H , -5,14 ) = KH ) 146.7 ) +31g -151+213-141=0

' HHG cost , cost ,- 2sint7

118 'H=pf2twtt4nt=4Tt4nF=1F=2FH±I=<Eacostitscost ,

- int )HP 't

ttfzsnt ,- Hunt ,

- cost >

HTYHFysiitttysrittcoit = Test =f= ,

Di ,F¥,,=fFsmt ,

- tasnt ,- cost >

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Quadric Surfaces

Equation Surface Equation Surface

Ellipsoid 2 2 2

2 2 2 1x y za b c

� �

Cone 2 2 2

2 2 2

x y za b c

�

Elliptic Paraboloid

2 2

2 2

x y za b c

�

Hyperboloid 1 Sheet

2 2 2

2 2 2 1x y za b c

� �

Hyperbolic Paraboloid

2 2

2 2

x y za b c

�

Hyperboloid 2 Sheets

2 2 2

2 2 2 1x y za b c

� � �

Curvature

� �� �

� � � �� � 3

' ' "

' '

T t r t r t

r t r tN

u

Definition: cosu v u v T�

sinu v u v Tu