Homework 5

# Homework 5 - Vincent(jmv — HW05 — Gilbert(57495 1 This...

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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. a-b m —8 correct 2. a-b m,_—10 3. 21-13 = “4,20 4. ab 3 —6 5. 21-13 = ~14 Explanation: The dot product, a. b, of vectors a = (o1,o2,o3), b r: (51,132,193) is deﬁned by a-b = a1b34—agbg—l—a3b3. Consequently, when a m (~2,~1,3>, b m (—2,3,~3>, we see that a-bzm8. 002 10.0 points Find thescalar projection of b onto a when bm2i+2j+8k, am2i~j—2k, 1. scalar projection H i Casi-q 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, a-bzl, |a|2=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 = w3i-lu6j-i-2k. 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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Homework 5 - Vincent(jmv — HW05 — Gilbert(57495 1 This...

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