• HW for 3/18

    Posted by Allen Olsen on 3/18/2019

    Objective: Use vectors to solve geometric problems.

    HW: This is a problem that uses the same technique that we used to solve the parallelogram problem in class.  We wish to show that, in any triangle, the line segments drawn from each vertex to the midpoint of the side opposite (the "medians"), all meet in one point R.

    1.a. Draw a triangle ABC, and label the vertices A, B, and C.  Let D and E be the midpoints of BC and AC, respectively.  Mark them on your diagram.  Draw BE and AD.

    b. Now let the vector S = AB, and T = AC.  Show that BC = T - S.

    Show that AE = (1/2) T, AD =(1/2) (S + T), and BE = (1/2) T - S.

    c. Let the point of intersection of AD and BE be point R.  Mark it on your diagram.

    d. Let AR = x AD, and let BR = y BE, where x and y are scalar constants.  What is the geometric meaning of these equations?

    e. Show that the equations in part (d) lead to the system of equations

        (1/2) x = 1 - y;

        (1/2) x = y.

    f. Solve these equations for x and y.

    2. a. In the same diagram for question 1, draw a line from C through R, meeting AB at point F.  Update your diagram.

    b. Let CF = a CR, and let AF = b S, where a and b are scalar constants.  What is the geometric meaning of these equations?

    c. Show that CR = (1/3) S - (2/3) T.

    d. Show that T + CF = AF, and that this leads to the equations

         1 - 2a/3 = 0;

         a/3 = b.

    e. Solve these equations for a and b.

    f. Why does this show that the three medians of any triangle meet in one point R?

     

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  • HW for 3/14 (A) and 3/15 (D)

    Posted by Allen Olsen on 3/14/2019

    HW: Enjoy your weekend!  I will be checking to see if you have done this homework on Monday!

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  • HW for 3/13

    Posted by Allen Olsen on 3/13/2019

    Objective: Summarize your work on vectors and dot products so far by writing up proofs of some of the major results.

    HW: Sect. 6.2, p. 465:

    33 - 38; 57, 58, 60.

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  • HW for 3/12

    Posted by Allen Olsen on 3/12/2019

    Objective: Use dot products to find projections of vectors.

    HW: First, some of you did not do the problems I posted yesterday.  Please look at the website and do them.

    Second, here is today's homework:

    Read Sect. 6.2 about projections.

    Then, on pp. 464 - 65:

    25 - 30.  Make sure to use the formula for projections in Section 6.2 on page 462.

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  • HW for 3/11

    Posted by Allen Olsen on 3/11/2019

    Objective:  Calculate and explain the geometric significance of dot products of vectors.

    HW:  Things to note: 1.  the vector i = <1,0> and the vector j = <0, 1>.  2. The word orthogonal means perpendicular.

    Sect. 6.2, pp. 464 - 66:

    1 - 7 (odd), 9 -18; 23, 24.

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  • HW for 3/7 (A), 3/8 (D)

    Posted by Allen Olsen on 3/7/2019

    Objective: Draw, stretch, add and subtract vectors.

    HW: Sect. 6.1, pp. 456 - 58:

    1 - 20.

    Also, locate problems similar to problems you got wrong in the Review Packet for the recent MidYear Test and report back to me whether there was such a problem, whether you did it, how completely you did it, and whether you got it right. I will be speaking with each of you about what you found next week.

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  • HW for 2/28 (A), 3/1 (D)

    Posted by Allen Olsen on 2/28/2019

    Objective: Prepare for Chapter 5 Quiz on Monday in class.

    HW: Write up a study guide for the work we have done on Trig Identities over the last few weeks.  In particular, you should remember the derivation of the formulas below.

    1. Sine double angle.  sin 2x = sin (x+x) = sin x cos x + cos x sin x = 2 sin x cos x.  (Uses sine addition formula).

    2. Cosine double angle.  cos 2x = cos (x+x) = cos^2 x - sin^2 x.    (Uses cosine addition formula).

    3. Other forms of cosine double angle, using a Pythagorean identity.

    a. cos 2x = cos^2 x - sin^2 x = cos^2 x - (1 - cos^2 x) = 2 cos^2 x - 1.

    b. cos 2x = cos^2 x - sin^2 x = (1 - sin^2 x) - sin^2 x = 1 - 2 sin^2 x.

    4.

    a. sin^2 (x/2) = 1/2 - (1/2 - (sin^2 (x/2)) ) = 1/2 - (1 - (2 sin^ (x/2)) )/2 = 1/2 - (cos (2 (x/2)) )/2 = (1 - cos x )/2

    b. cos^2 (x/2) = (2 cos^2 (x/2)) / 2 = 1/2 + (2 cos^2 (x/2) - 1) / 2 =1/2 + (cos 2(x/2)) / 2 = (1 + cos x)/2

    You will be asked to solve SSS, SAS, SSA, ASA, or AAS triangles using the Law of Sines and Law of Cosines, which you should memorize or know how to derive.

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  • HW for 2/27

    Posted by Allen Olsen on 2/27/2019

    Obj: Consolidate your knowledge of the derivation and usage of trigonometric identities.

    HW: Ch. 5 Review Exercises, pp. 443 - 46:

    3, 4, 6, 7, 12, 21, 25, 27, 29 - 34, 51 - 52, 54 - 58.

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  • HW for 2/26

    Posted by Allen Olsen on 2/26/2019

    Obj: Use the Law of cosines to solve SSS and SAS problems.  Be able to solve the "ambiguous case" in SSA, using the Law of Sines.

    HW: Sect. 5.5, p. 432:  19 - 22; 27 - 35 (odd);

    Sect. 5.6, p. 441: 5 - 8, 11, 13, 14, 16.

     

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  • HW for 2/25

    Posted by Allen Olsen on 2/25/2019

    Objective: Use the Law of Sines to solve triangle problems, given three parts.

    HW: Sect. 5.5, p. 432:

    5 - 12.

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