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Transcription

1 MATH 0 Exam (Version ) Solutions Setember, 00 S. F. Ellermeyer Name Instructions. Your work on this exam will be raded accordin to two criteria: mathematical correctness clarity of resentation. In other words, you must know what you are doin (mathematically) you must also exress yourself clearly. In articular, write answers to uestions usin correct notation usin comlete sentences where aroriate. Also, you must suly su cient detail in your solutions (relevant calculations, written exlanations of why you are doin these calculations, etc.). It is not su cient to just write down an answer with no exlanation of how you arrived at that answer. As a rule of thumb, the harder that I have to work to interret what you are tryin to say, the less credit you will et. You may use your calculator but you may not use any books or notes.. Show how to nd the distance between the oints P 0 (4; ; ) P (; ; 4). Solution: The distance between these oints is! P 0 P = ( 4) + ( ( )) + (4 ) = ( ) =. (a) Find the unit vector obtained by rotatin the vector h0; i throuh an anle of 0 counterclockwise about the oriin. (b) Exress the vector v = i + j as the roduct of its manitude direction. 0 i + j A. {z } {z } manitude direction Solutions: For art a, notice that the vector h0; i is already a unit vector that oints to the north (in the xy lane). If we rotate this vector 0 counterclockwise, we et a unit vector that oints 0 south of west. Based on out knowlede of the unit circle, D this vector is ; E. For art b, we see that the manitude of the iven vector is s 66 jvj = + = Thus the direction of this vector is jvj v = i + 66 j = i + j Therefore 0 66 v = B i + jc } A. {z } manitude direction

2 . Let u v be the vectors u = i + j v = i + j + k let be the anle between u v. (a) Find u v, juj, jvj. (b) Find cos (). (c) Find the rojection of u onto v (denoted by roj v u). (d) Find the scalar comonent of u in the direction of v. Solution: u v = = = i + j i + j + k + + (0) () juj = r + = Thus jvj = r + + =. cos () = u v juj jvj = = roj v u = u v jvj v = 9, i + j + k, the scalar comonent of u in the direction of v is u v jvj =.. Which of the followin statements are always true which are not always true? (Circle the correct choice.) (a) jvj = v v (always true, not always true) (b) u v = (v u) (always true, not always true) (c) (u) v = (u v) (always true, not always true) (d) u 0 = 0 (always true, not always true) (e) u ( u) = 0 (always true, not always true)

3 (f) u v = (v u) (always true, not always true) () u (v w) = (u v) w (always true, not always true) (h) (u v) v = ju vj jvj (always true, not always true) (i) ju vj = juj jvj cos () where is the anle between u v. not always true) (always true, (j) u v = juj jvj cos () where is the anle between u v. (always true, not always true) Gradin of this uestion will be as follows: Number correct Points Find either arametric euations of symmetric euations of the line that contains the oint P 0 ( 4; ; 4) is erendicular to the lane x + y + z =. Solution: The vector h; ; i is erendicular to the iven lane is thus arallel to the line in uestion. Hence this is a direction vector for the line in uestion. A vector euation for the line is Parametric euations are hx; y; zi = h 4; ; 4i + t h; ; i. x = 4 + t y = + t z = 4 + t < t <.. The osition vector of a article movin in sace is iven by r (t) = (t + ) i + t j + tk. (a) Find the velocity vector, v (t). (b) Find the acceleration vector, a (t). (c) Find the seed, v (t). (d) Find arametric euations for the line that is tanent to the curve of motion at the oint on the curve corresondin to time t =. Solution: v (t) = dr dt v (t) = jv (t)j = = i + tj + k a (t) = dv dt = j + (t) + = 4t +.

4 At time t = we have v () = i + j + k this vector is tanent to the curve of motion at time t =. Also the oint on the curve of motion that corresonds to time t = is (; 0; ). Thus the tanent line at this oint is iven by the arametric euations x = + t y = 0 + t z = + t < t <. 6. Please be as detailed as ossible in answerin this uestion (so that I can follow your reasonin easily): A rojectile is red with an initial seed of 00 m = s at an anle of elevation of 4. (a) Bein with the acceleration function a (t) = j (where = 9:8 m = s ) show the stes involved in obtainin the velocity function, v (t), the osition function, r (t). (b) Show how to determine when how far away (from where it is red) the rojectile will strike the round. Solution: Since then Also thus Therefore a (t) = j, v (t) = tj + v (0). v (0) = 00 cos (4 ) i + 00 sin (4 ) j = 0 i + 0 j v (t) = 0 i + r (t) = 0 ti + since r (0) = 0 we obtain r (t) = 0 ti + 0 t 0 t j. 0 t t j + r (0) t j. In arametric form, the x y comonents of the ath of motion are x (t) = 0 t y (t) = 0 t 0 t T f t where T f is the time that the rojectile ls. 4

5 To nd T f, we solve y (t) = 0 for t: Settin 0 t we obtain either t = 0 (obviously not T f ) t = 0, or t = 0 t = T f = 00 7 seconds. The rane of the rojectile is x (T f ) = 0 T f = 0 00! = 00 ; 0 meters (or about : kilometers).