assign2 - Math 136 Assignment 2 Due Wednesday Jan 20th 1...

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Unformatted text preview: Math 136 Assignment 2 Due: Wednesday, Jan 20th 1. Determine a) proj<2y1,1)(3,2, 1) and perp(2)1,1)(3,2,1). b) proj(-1)170’2)(2, —2, 1, 1) and perp(_171,0,2)(2,—2,1, 1). 2. Use a projection to find the distance from the point to the plane. a) The point P(2, —3, —1) and the plane 2:131 —— 3x2 — 5x3 : 7. b) The point P(2, 2, 3) and the plane 2x1 + $2 — 4mg 2 5. 3. Calculate the following cross-products. a) (2,1,2) >< (1,2,3). b) (1,2,3) >< (2,1,2). 4. Consider the parallelogram determined by the vectors 6 2 (1, 4) and 21 = (2, 3). Find the area of the parallelogram by a) using the 2 x 2 determinant. b) putting the vectors into R3 and using the cross—product. 5. Determine the volume of the parallelepiped determined by (1,1,4), (1, 3, 4), and (—2, 1, —5). 6. Let c? E R3. Prove that projd(perpa(f)) = 6 for any a? E R3. 7. Consider the following statement “If 55 X 17 = :3 x 732 with :3 ;£ 6, then 17 2 11 ”. Give a proof if the statement is true or a counterexample if the statement is false. Use MATLAB to complete the following questions. You do not need to submit a printout of your work. Simply use MATLAB to solve the problems, and submit written answers to the questions along with the rest of your assignment. ———————-———_._._._.____.______________ Linear Combinations and Properties of Vectors Review the posted MATLAB Introduction before attempting Questions 1 and 2 below. (See the course webpage in UW—ACE under Content -—> MATLAB.) In particular, 0 review how to enter vectors in MATLAB, 0 figure out how to add vectors together, and a figure out how to calculate a scalar multiple of a vector. dot To find the dot product of two vectors in MATLAB, use the dot command. For example, the dot product of vectors a = (4, 3, —1) and b = (—2, 5, 3) can be found as follows: >> a = [4; 3; -1] >> b = [-2; 5; 3] >> dotCa, b) MATLAB returns that the dot product is 4. IlOI'IIl To find the length of a vector in MATLAB7 use the norm command. For example, the length of a from the previous example can be found as follows: >> norm(a) MATLAB returns that the length of a is 5.0990. —-———-———-——————————_—_—_—_—_—_—_— -——-——-———-————————____—____ Consider the set of vectors {111,212, . . . ,v7} in R10 below: 121 = (9,—5,2,0,7,5,—1,—9,6,——1) v2 = (O, —2, —1,0, 3, 3,4, —2,4, 1) v3 (4, —2, —1,0,3,3,4, —5,8, 1) v4 = (7,5,12,5,—5,—3,—1,6,——8,4) v5 — (—1,0,1,0,—1,0,0,0,1,1) v6 = (0,4,2,0,—6,—6,—8,4,—8,—2) v7 — (2,678,5, —6, —2,8,8, —2, 7) Question 1 Enter the vectors 111,112, . . . ,1», into MATLAB and then find the following linear combinations: (a) azvl+v2+v3+v4+v5+v6+v7 (b) b=5U4—9’l}7 (c) c=2v1—3U3+3v5+v6—v7 Question 2 Find the angle, 6, in radians and degrees, between vectors v2 and v3. Hints: 1. Review the formula for c086 in the textbook, Section 1—2, p. 25. 2. Type help acos and help acosd at the prompt and read the documentation. ...
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