Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.ICN Textbook: Calculus Early Vectors Preliminary Edition By Stewart. Wednesday, March 26 11.3 Cross Product p672: 1,2,(4),7,10,(12),(16),17,19,20,23,(26),27,28,31 Main Idea: The determinant is a logical thing to compute. It should be good for something. Key Words: Cross Product, Area, Torque, Right Hand System Goal: Learn to use the cross product effectively --------------------------------------------------------------- Previous Assignment Page 664 Problem 28 Find correct to the nearest degree, the three angles of the triangle with the given vertices. P = ( 0,-1, 6) Q = ( 2, 1,-3) R = ( 5, 4, 2) P = ( 0,-1, 6) /\ / \__ / |\ __/ \ (-5,-5, 4) (-2,-2,9) /| \ Sqrt[89] / \ Sqrt[66] / \ / \ / \ / \ /_____________\______\ Q = ( 2, 1,-3) / R = ( 5, 4, 2) (3,3,5) Sqrt[43] 56 Cos[P] = ------------------ P = 43.0574 degrees Sqrt[66] Sqrt[89] 33 Cos[Q] = ------------------ Q = 57.7619 degrees Sqrt[89] Sqrt[43] 10 Cos[R] = -------------------- R = 79.1807 degrees Sqrt[66] Sqrt[43] check: P+Q+R = 180. degrees. --------------------------------------------------------- a = Graphics3D[Line[{ {2,1,-3},{5,4,2},{0,-2,6},{2,1,-3}}]]; b = Show[a,PlotLabel->"Page 664 Problem 28"]; Display["28.ps",b]; c = Show[a,PlotLabel->"Page 664 Problem 28", ViewPoint->({7/3,4/5,5/4}+{-10,10,0})]; Display["28x.ps",c]; -------------------------------------------------------------- Page 664 Problem 34 Find the values of x such that the given vectors are orthogonal. (x,x,-1),(1,x,6) x+x^2-6 = 0 x^2 + x -6 = 0 (x-2)(x+3) = 0 x = 2, x = -3. (2,2,-1)o(1,2,6) = 0 and (-3,-3,-1)o(1,-3,6) = 0. ------------------------------------------------- a = Graphics3D[ Line[{{0,0,0},{2,2,-1}}]]; b = Graphics3D[ Line[{{0,0,0},{1,2, 6}}]]; ab = Show[a,b,PlotLabel->"Page 664 Problem 34 x = 2"]; abx = Show[a,b,PlotLabel->"Page 664 Problem 34 x = 2",ViewPoint->{14,-13,2}]; Display["34.ps",ab]; Display["34x.ps",abx]; c = Graphics3D[ Line[{{0,0,0},{-3,-3,-1}}]]; d = Graphics3D[ Line[{{0,0,0},{ 1,-3, 6}}]]; cd = Show[c,d,PlotLabel->"Page 664 Problem 34 x = -3"]; cdz = Show[c,d,PlotLabel->"Page 664 Problem 34 x = -3",ViewPoint->{-21,17,12}] Display["34y.ps",cd]; Display["34z.ps",cdz]; --------------------------------------------------------------------- Page 664 Problem 58 If r = (x,y,z) and a = (a1,a2,a3) and b = (b1,b2,b3), show that the vector equation (r-a).(r-b) = 0 represents a sphere and find its center and radius. . ''' . . ' . \/ .`. . . ' '. . . . ' '.. a-----------------------b The angle inscribed in a semicircle is a right angle. The center is the midpoint of the line segment ab and the distance between a and b is the diameter. (x-a1)(x-b1)+(y-a2)(y-b2)+(z-a3)(z-b3) = 0 x^2 - (a1+b1)x + y^2 - (a2+b2)y + z^2 - (a3+b3)z + a1 b1 + a2 b2 + a3 b3 = 0 x^2 - (a1+b1)x + 1/4 (a1+b1)^2 + y^2 - (a2+b2)y + 1/4 (a2+b2)^2 + z^2 - (a3+b3)z + 1/4 (a3+b3)^2 + a1 b1 + a2 b2 + a3 b3 = 1/4 (a1+b1)^2+ 1/4 (a2+b2)^2+ 1/4 (a3+b3)^2 (x-(a1+b1)/2)^2 + (y-(a2+b2)/2)^2 + (z-(a3+b3)/2)^2 = (a1-b1)^2 + (a2-b2)^2 + (a3-b3)^2 --------------------------------- 4 Center is at the midpoint of the segment from a to b and the radius is half the distance from a to b. ------------------------------------------------------------------- Page 664 Problem 60 Find the angle between a diagonal of a cube and a diagonal of one of its faces. ______________________________ /| /| / / | / | / | / / | / | / | / / | /_____________________________/. | | | . ' | . | | |. ' | .| | | | . '\ | __| | | . ' | . | | | . ' | .| | |_'_ _ _ _ _ _ _ _ _ _|_ _ _ | | / | / | | / | / | / | | / | / | / |/____________________________|/ Main diagonal = (-1,-1,-1). Right end facial diagonal (-1,0,-1) 2 Cos[theta] = -------------- = Sqrt[2/3] = 0.816497 Sqrt[3] Sqrt[2] theta = 35.2644 degrees. ---------------------------------------------------------------- New Material: Determinants ============ | 1 2 3 | | 2 0 1 | = | 3 1 1 | Expansion Rule for 3x3 determinant. | 1 2 3 | 1 2 | 2 0 1 | 2 0 = 1 0 1 + 2 1 3 + 3 2 1 -(3 0 3 + 1 1 1 + 1 2 2) | 3 1 1 | 3 1 = 0+6+6-(0+1+4) = 7 Results about determinants. 1. You can factor a number out of any row or column and place it outside the determinant. 2. If you interchange two rows or two columns, you change the sign of the determinant. 3. The determinant is the volume of the parallelopiped. ___________ /| .| / |_________| / / . / / / . / / W/ . / | U | / / . / volume = | V | /../....... / | W | U | / ./ |/________/ V 4. Cramers rule. x + y - z = 10 2x + 3y + 2z = 13 y + 7z = 24 x = Dx/D, y = Dy/D, z = Dz/D. | 1 1 -1 | D = | 2 3 2 | | 0 1 7 | |10 1 -1 | Dx= |13 3 2 | |24 1 7 | | 1 10 -1 | Dy= | 2 13 2 | | 0 24 7 | | 1 1 10 | Dz= | 2 3 13 | | 0 1 24 | | a b 1 | The area of a triangle (a,b),(c,d),(e,f) = 1/2 | c d 1 | | e f 1 | So one would think that the determinant expansion could play a useful part in vector calculations. | i j k | Definition (a,b,c) x (d,e,f) = | a b c | = (bf-cd,-ab+cd,ae-bd) | d e f | This vector has the following properties. 1. AxB = -BxA. 2. AxB is perpendicular to both A and B and forms a right handed system. 3. |AxB| is the area of the parallelogram formed by A and B. 4. |AxB| = 1/2 |A||B| Sin[t] where t is the angle between A and B. 5. (AxB).C = the volume of the parallelopiped formed by A,B,C. Applications: 1. Find the distance from (1,2,3) to the line through (2,1,0) and (3,1,1); (1,2,3) _/| /|| A=(-1,1,3)/ | |A| Sin[t] / | / | /_t____________________\__ (2,1,0) B=(1,0,1) /(3,1,1) | i j k | |AxB| |-1 1 3 | |(1,4,-1)| Sqrt[18]/Sqrt[2] = 3 Take ------ = | 1 0 1 | = ---------- = |B| ----------- Sqrt[2] Sqrt[2] 2. Find the distance from (2,2,4) to the line through (2,0,1) and (1,1,1). 3. Find the distance between the facial diagonals of a Cube 4. Find the volume of the parallelopiped generated by the vectors from (0,0,0) to (1,2,0), (1,1,1), and (2,1,0. 5. Find the area of the triangle with vertices (1,4,6), (-2,5,-1) and (1,-1,1) 6. Complete the following table U D N S E W ------------------------------ U | | | | | | | --------------------------------- D | | | | | | | --------------------------------- N | | | | | | | --------------------------------- S | | | | | | | --------------------------------- E | | | | | | | --------------------------------- W | | | | | | | --------------------------------- 7. Graphing a sphere, we can identify north pole and up. How will we identity the local coordinates of N,S,E,W.