(10/14/08)Math 10C. Lecture Examples.Sections 13.1 and 13.2. Vectors†Example 1 Find the x- and y-components of the vector u of length 10 with angle ofinclination56π.Answer: Figure A1a • u = h−5√3, 5i • Figure A1bxyuθ =56π10xyu = h−5√3, 5i5−5√3θ =56π10Figure A1a Figure A1bExample 2 Find an ang le of inclination of the vector w= h3, 4i.Answer: Figure A2. • θ = tan–1(43)x3y4w = h3, 4iθFigure A2†Lecture notes to accompany Sections 13.1 and 13.2 of Calculus by Hughes-Hallett et al.1Math 10C. Lecture Examples. (10/14/08) Sections 13.1 and 13.2, p. 2Example 3 Calculate (a) v + w and (b) w −v for v= h 4, 1i and w = h1, 3i. Then drawthe four vectors.Answer: (a) v + w = h5, 4i • Figure A3a (b) w − v = h−3, 2i • Figure A3bx2 4 6y1234vwv + wx2 4 6y1234vww − vFigure A3a Figure A3bExample 4 Write 3h4, −1i − 2h10, −5i in the form ha, bi.Answer: 3h4, −1i − 2h10, −5i = h−8, 7iExample 5 Give the unit vector evand the vector w of length 5 with the same directionas v= h−3, 2i.Answer: ev=h−3, 2i√13• w =h−15, 10i√13Example 6 Express 3(4 i − j) − 2(10i − 5j) in the form ai + bj.Answer: 3(4i − j) − 2(10i − 5j) = −8i + 7j. (Ex ample 4 is the s ame calculation with different notation.)Example 7 One man is lifting a boulder with a rod while another is pulling it with arope as in Figure 1. (a) Find the x- and y-components of the two forcevectors, with t he usual orientation of axes. (b) Find the resultant of thetwo forces and the approximate decimal values of its magnit ude and angleof inclinationFIGURE 1Answer: (a) [Force exerted by the man with the rod] = F = 300cos718π, sin718πpounds. •[Force exerted by the man with the rope] = G = 150cos19π, s in19πpounds.(b) [Resultant] = h300 cos718π+ 150 cos19π, 300 sin718π+ 150 sin19πi.= h244, 3 33i pounds •[Magnitude of the combined force.= 413 pounds • [Angle of inclination].= ta n–1333244.= 0. 94 radians.Sections 13.1 and 13.2, p. 3 Math 10C. Lecture Examples. (10/14/08)Example 8 Write z = u + 2v + 3w in the form a i + b j + c k, where u = 3 i − j, v = j − 3 kand w = i + k.Answer: z = 6 i + j − 3 kExample 9 Three adjacent vertices of a parallelogram PQRS in space are P = (1, 3, 2),Q = (4, 5, 3), and R = (2, −1, 0). What are the coordinates of the point Sopposite Q?Answer: Use the schematic sketch in Figure A9. • S = (−1, −3, −1)Figure A9Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 12.1: Examples 1, 2, 6, 7Section 12.3: Examples 1, 2, 6‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript an d not to the chapters and sectionsof the t ex tbook for the
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