Ubc Math 200 Midterm 1A 2011

Math 200 sec. 103 (Bond) Exam1 (100 pts) Name: _

You have 50 minutes. Please show all work to get full credit.

1. (20 pts, 5 per part)u = <1 2 3> v = <-1 -1 5>'" , , .

a. Find lul and Ivl.

~j r'f~f£J.® J1~1f25 =\~8[iJrz~(+t-(L)H)+(jy» >-1-2-f15{3J ..c. Find the angle between u and v.

~~~~ I~llvl(oft

..0 20)-I U~v ...,1}J (oj li$f - -.I (0) IU~jd. Find uxv .

u-xV ::

...:;:. ~ ~

\ ) k.I 2 3-\ --~5

.; CQ)(S) -(3)(--1ilt - [(1)(')) -G)(-OJS

-t [ (I) (-1) - 0) HJ]f=- [lot- ~It - ['3-+31~T [-I+J-j

...> ->'.~ ~~'((V::: \) l -...,'6) +- ~

•••• ...;>

...• ~lA,'>I:V (1\ -Y,( I)

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Math 200 sec. 103 (Bond) Exam} (100 pts) Name: _

2(25 pts): For this problem, A = (1,3,4) B = (3,3,1) C = (1,4,-3)

a) Find the area of the triangle whose corners are located at the pointsA, B, and C.

4~<2-0-3~!\ I / '/

,j(c =- <~JI/-t>

Aix~ c: (; J-f) =- 3~ +IY") +2~~ 0 I -+ ::: L;, I,2> ..

Find a non-zerolv) ogonal to the plane which contains

the three points A, B, and C.

\t) ftS.----

f;v

A rea -.:: , 1.-A ~Ij ABxjrc

~J~yJ~~ 5,1)

..::--l

,6ft),~ ~

r) =- AJ) x#:- =: 0,IQ/ :2>

c) Find an equation for the plane containing the three points A, B, andC.

-.:l ~11) 6 AP z: 0 I lAa--t- ~S)

< 3 I 1'1/1),,\>( -1!j -\ ? -4) -:::0---( 3(~-i) +h(~-3) ,})C2-t)~

~

. tr.

~::: (K;3?) '11\ -th flOYU-;

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Math 200 sec. 103 (Bond) Exam1 (100 pts) Name: _

® T'{jO ~ \ I'lh '. i.c:. (D -!:- () '\ f. 1 : v= ( -j 2X +') ~ --1=-1 ;i).---------- '>I SI ) . -

f - (J2- (I J) -- >( ~ j -+ 3 - J -~K~

I - SI 5; I;j :::.II <C

L II 9 - I \ x:--' I-: (t) ~ (1"~;)OPot tW,::: Q -t:) Z ~/-'>})) + t (ir3); . -5" - 5

.c----- ----

....-l ~

f\\~ ~l

2-3 -)--1

@( P~'~1-d-- \v~'t)d+~)

3. (25 pts) Consider two planes: Plane 1 has the equation2x + 3y -7z = 4, and Plane 2 has the equation x - y + 3z = 3.

a. (1Opts) Find the angle between the two planes.

if, = ()} I-f) (n, z: ,(\ -I) .)>-' ~ . il _ :=JEl~~

-;: ['1[\ InJ} (D.>G- 1 = ! ~ l·r--~ ~ ~ - Cd> (-22. , - V b)

- iLJ\T (G/8- - l~·b. (15pts) Find a para1netric equation tr the line where

Plane 1 intersects Plane 2.

I~h~::r"2-V;--

["'·ll ~~ ~ ~- \) -:: .. ~V :: Yll ~ ~ ~ - 1~--1 ~ II - \-3j --t; ~\ -- \ .5.--

f~L ~ lo:f) ~ ~ 2X-.¥3t(/'-{ 2(;jB)Hj 2 1{-

( x -'F-3~~. I <y +i o=-~~ \vI-r--3

x~J L~:= - 3,

A-~ f-~ ~ 135~ -S

f e::: (~-~ 0)/ $)

Of. - f ~~

< ~-=.l-0> + / 1 -) 3 - 5>-{:I >1 "I)

; --------~(-t)=-

Math 200 sec. 103 (Bond) Examl (100 pts) Name: _

4. (20pts) What is the shortest distance from the point (1,2,-3) tothe plane II whose equation is x - 3y + Z = 2? (0 tJ )

~ \:>0 ~ )?~~,le-s

p("O~~ (W)~

~ -0 \~~

r\\\ .

~jI)

p

-\0---~'~-I-I-~ 10---V1T

?(,)..~

fD P ~ <-0 2/ --S>~

//J -==- < 1)~3/1 )~ ~Rr f Y':: 1~-5

-=- - rD

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r; Math 200 sec. 103 (Bond) Examl (100 pts) Name: _

5. (1Opts) Consider the line A with the following parametricequations:

x = 2 -- 2t Y= 3 + 3t z = -5 -- tWhere does the line A intersect the plane 2x-y+z=4?

f\i'\~{ :

ll;-A) = 0~~t)1- (-S-t) ~i\ __~+ ~)- 3t -S ~ t- - ~

---Z~ - f c cr ,t- =-- -I

~

~~i ~:: V l =-~

f -= lLfIO / -~)

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