Download Test 3 for Calculus III - Spring 2006 | MTH 253 and more Exams Advanced Calculus in PDF only on Docsity!
MTH 253 - Spring Term 2006
Test 3 Name
Please feel free to use your calculator for any and all calculations.
- Write into each provided blank the requested object. No work need be shown. (2 points each)
a. What is 3, โ 5, 2 โ
5, โ 3, โ 7? a.
b. What is 8, 2, โ 4 ร โ2, โ 1,3? b.
c. What is โ 5 i ห^ + ห j โ k ห? c.
d. What is โ12, 4, 0 โ 3, โ 3, โ 4? d.
e. What is the unit vector that points in the same e. direction as the vector 3, 0, โ 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
โ 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 โ 12 k ห?
- 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^ โ 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. resultantโs 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
- Suppose that the vector u^ G^ = 3,1, โ 2 were drawn with its tail at the head of the
vector v^ G^ = 5, โ1, 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)
- 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
- Convince me that the line 2 4 1 3 5
x โ (^) = y + = z โ lies entirely in the plane 3 x โ 4 y โ z = 21 (as
illustrated in Figure 4). (6 points)
Figure 4
- Find the equation of the plane perpendicular to 3 x โ 4 y โ z = 21 that contains the line 2 4 1 3 5
x โ (^) = y + = z โ. (Note that this relates back to question 8.) Show all relevant work in a
well-organized and well-documented manner. (8 points)
- Suppose that the line, l, passes through the points (^) ( 2, โ 5, 4)and (^) ( โ1, โ 1, 4).
b. Suppose that a line, l 1 , perpendicular to l, passes through the point (^) ( โ80, 21, โ (^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 hasnโt shown up since page 2?) (10 points)
