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Unformatted text preview: Vincent (jmv???) — HW05 — Gilbert  (57495) 1 This print—out should have 12 questions.
Multiplemchoice questions may continue on
the next column or page u find all choices
before answering. 001 10.0 points Determine the dot product of the vectors a meg—1,3), b m (—2,3,—3). 1. ab m —8 correct
2. ab m,_—10 3. 2113 = “4,20 4. ab 3 —6 5. 2113 = ~14
Explanation: The dot product, a. b, of vectors
a = (o1,o2,o3), b r: (51,132,193)
is deﬁned by
ab = a1b34—agbg—l—a3b3.
Consequently, when
a m (~2,~1,3>, b m (—2,3,~3>, we see that abzm8. 002 10.0 points Find thescalar projection of b onto a when bm2i+2j+8k, am2i~j—2k, 1. scalar projection H
i Casiq 0310“: 2. scalar projection 3. scalar projection m ——2
. . Ki 4. scalar prejectlon 2 Mg correct
. . 8 5. scalar projection 2: ———g Explanation: _
The scalar projection of b onto a is given in
terms of the dot product by a b
b = ——.
compa Ia}
Now when
b:2i+2j+3k, a=2i—— j—Zk,
we see that . r
ab = +4. la! = «(2)2 +<1)2 +~ e2)?
Consequently,
4
compab = mg
keywords: 003 10.0 points Find the vector projection of b onto a when b=(——4,3), am(~3,1).
1. vector proj. m j—%(~4, 3)
2. vector proj. = %(W4, 3,) 17 3. vector proj. m ﬁ(~3, l)
4. vector proj. 2 %(WS, 1)
5. vector proj. m g<~4, 3)
6. vector proj. = 3(w3, I) correct Vincent (jmv???) — HW05  Gilbert — (57495) 3 Consequently, The vector projeCtion of a vector it) onto a
vector 21 is given in terms of the dot product axb m<3,3,—4> . by ab
_ projab = ( )a. iai2 "WWW On the other hand, since the unit cube has
keywords: vectors, cross product Side_§ength I? 006 10.0 points A = (0,0,1), B : (1, 0, 0), While D m (0, 1,0). In this case E is a The box Shown in directed line segment determining the vector a m (0,1,—1)=jw~k,
while Hi determines the vector b : (1,0,w1) : imk.
For these choices of a and b, abzl, a2=2. Consequently, the vector projection of 21?
onto 271—1.?) is given by . 1 .
proiab = Eoﬁk)  is the unit cube having one corner at the
origin and the coordinate planes for three of m
its adjacent faces, keywords: vector projection, dot product, . ‘ . . unit cube com onent
Determine the vector prejectlon of E on ’ p ’ X13. 007 10.0 points ‘  1 . . .
1. vector projectmn m it} _ k) correct Find the vaiue of the determinant 1 2 —1
2. vector projection = wéﬁwk) D = 2 MB m1
1 —2 ——3 m1
3. vector ro'ection m — i—k
p J 2( ) 1. D m 20 correct
2
4. vector projection m —§(i+j—k) 2. D 2 18
2 3. D = 16
5. vector projection = §(i+jwk)
1 4. D m 14
6. vector projection : —— i—k
2( l 5. D = 12 Explanation: Explanation: Vincent (jmv777) w HWO5 m Gilbert — (57495) 5 2
6. v = i(% ° ~ g ' —— correct
Explanation:
The nonzero vectors orthogonal to a and b
are all of the form v = Alaxb), A%U, with A a scalar. The only unit vectors orthog—
onal to a, b are thus But for the given vectors a and b, i j k
a><b= 4 1 3
6 2 3
m 1 3. 4 3 . 4 1
“l2 31ml6 3l3+l6 2lk
= w3ilu6ji2k.
Inthiscese,
la><bl2=49. Consequently, keywords: vector product, cross product, unit
vector, orthogonal, 010 10.0 points Determine the length of the cross product
of a, b when lal = 5, lbl 2: 2 and the angle
between a, b is 7r/3. 1. length = 10 2. length m 0 H
g 3. length 4. length : 5\/§ correct
5. length = 5
Explanation: The length of a x b is given by
la >< bl 3: lal lbl 81119 where 6, O S 9 g 71", is the angle between a
and b. Consequently, length = 10sin(7r/3) = 5V5 keywords: vector, cross product, length, 011 18.0 points Compute the volume of the parallelepiped
determined by the vectors a m (4, —4, ms), b = (amt, m2), and
c = (1,4, m2). 1. volume = 19 20 correct 2. volume 3. volume m 23 21 4. volume 5. volume = 22 Explanation:
For the parallelepiped determined by vec—
tors a, b, and c its volume m la (b x c)l.
But
4 —4 —3
a(chlz'Z ——4 ——2
1 4 m2
—4 —2 2 m2 2 W4
= 4 +4 —— 3
4 «~2 1 W2 1 4 ...
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 Fall '09
 Gilbert
 Dot Product, vector projection, scalar projection

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