Mei Qin Chen citadel-math231 WeBWorK assignment number Homework2 is due : 09/02/2011 at 09:00am EDT. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information.

This file is /conf/snippets/setHeader.pg you can use it as a model for creating files which introduce each problem set.

The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA’s or your professor for help. Don’t spend a lot of time guessing – it’s not very efficient or effective. Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 2 ∧ 3 instead of 8, sin(3 ∗ pi/2)instead of -1, e ∧ (ln(2)) instead of 2, (2 +tan(3)) ∗ (4 − sin(5)) ∧ 6 − 7/8 instead of 27620.3413, etc. Here’s the list of the functions which WeBWorK understands. You can use the Feedback button on each problem page to send e-mail to the professors.

1. (1 pt) local/Library/Rochester/setVectors2DotProduct- ~u ·~v +~v ·~w = /UR VC 1 11.pg Correct Answers: Find a · b if ||a|| = 7, ||b|| = 2, and the angle between a and b is • 1 π − 10 radians. • <-4,2,-4> • 81 • -4 a · b = 5. (1 pt) Library/Michigan/Chap13Sec3/Q05.pg Correct Answers: Let ~a,~b,~c and~y be the three dimensional vectors • 13.3147912281321 ~a = 4˜j + 2k˜, ~b = −4˜i + 7˜j + 5k˜, ~c = 4˜i + 4˜j, ~y = 3˜i + 6˜j 2. (1 pt) local/Library/Rochester/setVectors2DotProduct- /UR VC 1 12.pg Perform the following operations on these vectors: (a)~c ·~a +~a ·~y = ~ If a = ¡4, -9, -5¿ and b = ¡4, -8, -2¿, (b) (~a · b)~a = find a · b = . (c) ((~c ·~c)~a) ·~a = Correct Answers: SOLUTION ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ • 98 (a)~c·~a+~a·~y = (4i+4j)·(4j+2k)+(4j+2k)·(3i+6j) = 16 + 24 = 40. 3. (1 pt) local/Library/Rochester/setVectors2DotProduct- (b) (~a ·~b)~a = ((4˜j + 2k˜) · (−4˜i + 7˜j + 5k˜))(4˜j + 2k˜) = /UR VC 1 13.pg (38)(4˜j + 2k˜) = 152˜j + 76k˜. (c) ((~c ·~c)~a) ·~a = (((4˜i + 4˜j) · (4˜i + 4˜j))(4˜j + 2k˜)) · (4˜j + What is the angle in radians between the vectors 2k˜) = ((32)(4˜j + 2k˜)) · (4˜j + 2k˜) = (32)(20) = 640. a = ¡-4, 3, 10¿ and Correct Answers: b = ¡-10, -4, -10¿? • 40 Angle: (radians) • 152j+76k Correct Answers: • 640 • 2.02436709925895 6. (1 pt) Library/272/setStewart12 3/problem 1.pg 4. (1 pt) Library/FortLewis/Calc3/13-3-Dot-product/HGM4-13-3-04- The-dot-product.pg Find a • b if Perform the following operations on the vectors ~u = a = h5,0,−2i and b = h3,2,3i and h−2,1,−2i,~v = h−2,0,1i, and ~w = h2,1,−2i.

~u ·~w = Is the angle between the vectors ”acute”, ”obtuse” or ”right”?

(~u ·~v)~u = Correct Answers: ((~w ·~w)~u) ·~u = • 9 1 • acute • Positive • Positive 7. (1 pt) Library/272/setStewart12 3/problem 2.pg • Negative • Zero • Negative Determine if the pairs of vectors below are ”parallel”, ”or- • Positive thogonal”, or ”neither”. • Negative a = h1,5,4i and b = h5,25,−65/2i are • Zero 9. (1 pt) Library/272/setStewart12 3/problem 4.pg a = h1,5,4i and b = h5,25,20i are Find the scalar and vector projection of the vector b = a = h1,5,4i and b = h−5,−25,−20i are h3,−1,5i onto the vector a = h−4,−1,−1i.

Scalar projection (i.e., component): Correct Answers: Vector projection h , , i • orthogonal Correct Answers: • parallel • parallel • -3.77123616632825 • 3.55555555555556 8. (1 pt) Library/FortLewis/Calc3/13-3-Dot-product/geometric-dot- • 0.888888888888889 product/geometric-dot-product.pg • 0.888888888888889 10. (1 pt) Library/272/setStewart12 3/problem 5.pg Several unit vectors ~r,~s,~t,~u,~n, and ~e in the xy-plane (not three-dimensional space) are shown in the figure. A constant force F = 0i + 2j − 2k moves an object along a Using the geometric definition of the dot product, are straight line from the point (2, 1, 1) to the point (-5, -3, -3). the following dot products positive, negative, or zero? Find the work done if the distance is measured in meters and You may assume that angles that look the same are the the force in newtons. Include units in your answer. (Note, units same. are case sensitive. Clicking on the link units will give a list of units.) ? 1. ~r ·~s ? 2. ~s ·~t Answer = Correct Answers: ? 3. ~e ·~r • 0 J ? 4. ~e ·~s 11. (1 pt) local/Library/Rochester/setVectors3CrossProduct- ~ ? 5. ~n ·t /ur vc 2 2.pg ? 6. ~t ·~u Let a = ¡3, 5, 6¿ and b = ¡5, 4, 3¿ be vectors. ? 7. ~r ·~u Compute the cross product a × b.¡ , , ¿ ? 8. ~n ·~e Correct Answers: • -9 • 21 • -13 12. (1 pt) Library/Rochester/setVectors3CrossProduct/ur vc 2 4.pg If a = i + 7j + k and b = i + 12j + k, find a unit vector with pos- itive first coordinate orthogonal to both a and b. i + j + k Correct Answers: • 0.707106781186547 (Click on graph to enlarge) • 0 Correct Answers: • -0.707106781186547

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