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Test 3 for Calculus III - Spring 2006 | MTH 253, Exams of Advanced Calculus

Material Type: Exam; Class: Calculus III; Subject: Math; University: Portland Community College; Term: Spring 2006;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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MTH 253 - Spring Term 2006
Test 3 Name
Please feel free to use your calculator for any and all calculations.
1. Write into each provided blank the requested object. No work need be shown. (2 points each)
a. What is
3, 5, 2 5, 3, 7โˆ’โ‹…โˆ’โˆ’? a.
b. What is
8,2, 4 2, 1,3โˆ’ร—โˆ’โˆ’ ? b.
c. What is ห†
ห†ห†
5ijkโˆ’+โˆ’? c.
d. What is 12,4, 0 3, 3, 4โˆ’โˆ’โˆ’โˆ’? d.
e. What is the unit vector that points in the same e.
direction as the vector 3,0, 4โˆ’?
f. What is
()
ห†ห† ห†
ij jร—ร—
? f.
g. What is
()
ห†
ห†ห†
ijkร—ร—
? g.
h. What is the vector that has twice the length of the h.
vector 1
2, ,7
2
โˆ’ and points in the opposite direction
as that vector?
i. What vector of length 2 points in the same i.
direction as the vector ห†
ห†
512jkโˆ’?
pf3
pf4
pf5
pf8
pf9
pfa

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MTH 253 - Spring Term 2006

Test 3 Name

Please feel free to use your calculator for any and all calculations.

  1. Write into each provided blank the requested object. No work need be shown. (2 points each)

a. What is 3, โˆ’ 5, 2 โ‹… 5, โˆ’ 3, โˆ’ 7? a.

b. What is 8, 2, โˆ’ 4 ร— โˆ’2, โˆ’ 1,3? b.

c. What is โˆ’ 5 i ห†^ + ห† j โˆ’ k ห†? c.

d. What is โˆ’12, 4, 0 โˆ’ 3, โˆ’ 3, โˆ’ 4? d.

e. What is the unit vector that points in the same e. direction as the vector 3, 0, โˆ’ 4?

f. What is ( i ห†^ ร— ห† j^ )ร— ห† j? f.

g. What is ( i ห†^ ร— ห† j ) ร— k ห†? g.

h. What is the vector that has twice the length of the h. vector 2, 1 , 7 2

โˆ’ and points in the opposite direction as that vector?

i. What vector of length 2 points in the same i. direction as the vector 5 ห† j โˆ’ 12 k ห†?

  1. Please write the requested information in the provided blanks with regards to the vectors u^ G^ and v^ G^ illustrated in Figure 1. (2 points each)

a. Which is greater, u^ G^ + v G^ or v^ G^? a.

b. Which is greater, u^ G^ โˆ’ v G^ or v^ G^? b.

c. Which is greater, proj u G^ v G^ or proj v G^ u G^? c.

d. Which is greater, comp u G^ v G^ or comp v G^ u G^? d.

e. In what direction does u^ G^ ร— v G^ point? e.

f. Which is true: u v^ G Gโ‹… < 0 , u v^ G Gโ‹… > 0 , or u v^ G^ โ‹… G^ = 0 f.

g. What vector should be added to u^ G^ so that the g. resultantโ€™s head lays at the midpoint of v^ G^?

h. If the vectors were both rotated 90O^ in the clockwise h. direction, in what direction would u^ G^ ร— v G^ point?

u^ G^

v^ G

Figure 1

  1. Suppose that the vector u^ G^ = 3,1, โˆ’ 2 were drawn with its tail at the head of the

vector v^ G^ = 5, โˆ’1, 0. What would be the smaller angle formed by the two vectors? Draw a picture of the situation and show all relevant work in a well-organized and well-documented manner. (10 points)

  1. The intersecting lines L 1 and L 2 lie on a common plane as illustrated in Figure 2. Find the equation of this plane. Show all relevant work in a well-organized and well-documented manner โ€“ this includes appropriate annotations on Figure 2. (8 points)

Figure 2

L 1 L 2

x t x t y t y t z t z t

  1. Convince me that the line 2 4 1 3 5

x โˆ’ (^) = y + = z โˆ’ lies entirely in the plane 3 x โˆ’ 4 y โˆ’ z = 21 (as

illustrated in Figure 4). (6 points)

Figure 4

  1. Find the equation of the plane perpendicular to 3 x โˆ’ 4 y โˆ’ z = 21 that contains the line 2 4 1 3 5

x โˆ’ (^) = y + = z โˆ’. (Note that this relates back to question 8.) Show all relevant work in a

well-organized and well-documented manner. (8 points)

  1. Suppose that the line, l, passes through the points (^) ( 2, โˆ’ 5, 4)and (^) ( โˆ’1, โˆ’ 1, 4).

b. Suppose that a line, l 1 , perpendicular to l, passes through the point (^) ( โˆ’80, 21, โˆ’ (^6) ). At what point do the two lines meet? Show all relevant work in a well-organized and well- documented manner โ€“ this includes an appropriately annotated diagram. (Hint: Is there any topic you projected would be on the test that hasnโ€™t shown up since page 2?) (10 points)