or 12 -12118,11 - new mexico institute of mining and...
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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
2
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
3
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 .
4
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 >
5
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
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