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Mathematics 1229A/B Lecture Notes - Lecture 6: Z-Dna

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22 Dec 2019

School

Western University

Department

Course

Mathematics 1229A/B

Professor

Mohsen Mollahajiaghaei

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MATH 1229A/B Full Course Notes

MATH 1229A/B Full Course Notes

Document Summary

Then n x p 0 can usepointg or r insteadof p showstheplaneit. Ex find an equationfortheplanedeterminedbythepoints pi cl0,0 qe lo l o and l o x c i o l l l thenormalvector n a f l. X t y t z e l standardform. Theusualformfortheequation of a line in r2 is ax by c. Howaboutr canwedescribe a linein rs ax 1byecz d this is theequationfor a plane nota line. Letuto bea vectorand let pbea pointwewanttodescribetheline through pand parallelto v theequationforthisline is x p tu orsince dependson 1 we write. I 2,37andparallel totherector thepoint ve l l 2 o ptv p. V g p v to pto ep t i pi v p v. Cl 2,3 t ec l 2,0 pointparallelpoint ta the magnitude the direction l t 2 2t 3. Let xlt p t tu to bea pointparallelfor a line wewriteit in terms of its coordinates letpe p paps vacu vaus and x e x y z then.

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Related Questions

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Let r in and u =(1 1 0), v = (1 4 1), w = (r2 1 r2), b = (3+2r 5+12r 2r). For which values r in is the set {u. v, w} linearly independent? For which values r in is the vector b a linear combination of u. v and w? For which of these values of r can b be written as a linear combination of u. v and w in more than one way? The set 3 of all column vectors of length three, with real entries, is a vector space. Is the subset B = {(x y z) in 3 | xy + yz = 0} a subspace of 3? Justify your answer. Show that the set of all twice differentiable functions f : satisfying the differential equation sin(x) f"(x) + x2f(x) = 0 is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f" denotes the second derivative of f. Let S be the following subset of the vector space 3 of all real polynomials p of degree at most 3: S = {p in 3 | p(l) = 0, p'(l) = 0}, where p' is the derivative of p. Determine whether S is a subspace of 3. Determine whether the polynomial q(x) = x - 2x2 + x3 is an element of S. u = (1 1 0), v = (1 4 1), w = (r2 1 r2) if {u, v, w} is linearly independent set 1(4r2 - 1)-1(r2 - r2) ne 0 4r2 ne 1 r ne plusmn1/2 this set is linearly independent set for values of e ne plusmn1/2 - - - - - - - (3+2r 5+12r 2r) = a1 (1 1 0) + a2 (1 4 1) + a3 (r2 1 r2) this forms system of equations in a1, a2, a3 if 4r2 - 1 ne 0 then system has unique solution if 4r2 - 1 = 0 and -2 - 4r = 0 rArr r = -1/2, the system has more than one solution if 4r2 - 1 ne 0 b can be written as linear combination of u,v,w as uniquely if r = -1/2 then b can be written as linear combination of u,v,w more than one way - - - - - - - - - (1 1 -1) in B and (1 0 1) in B (1 1 -1) + (1 0 1) = (2 1 0) not in B since 2times1+1times0 ne 0 so addition of vector is failed, this set is not forms subspace consider the following differential equation (sinx)f"(x) + x2f (x) = 0 each solution is continuous function the set of all solutions is subset of space of set of all continuous functions n from R to R let f,g are two solutions (sinx)f"(x) + x2f (x) = 0 (sinx)g"(x)+x2g(x) = 0 ccheck whether af+bg is solution or not (sinx)f"(x) + x2f(x) = 0 rArr (sinx) af"(x) + x2?f (x) = 0 (sinx)g"(x) + x2g (x) = 0 rArr (sinx)bg" (x) + x2bg (x) = 0 (sinx)(af"(x)+bg"(x)) +x2 (af (x) + bg(x)) = 0 af+bg is also solution to differential equation the set of all solutions to differential equation forms subspace - - - - - - - - - - consider the following subset of P3 {p in P3/.p(l)=0, p'(l) = 0} let p, q in S rArr p(l)= 0, p'(l) = 0 and q(l) = 0, q'(l) = 0 (ap + bq) (l) = ap (l) +b q(l) = a (0) + b (0) = 0 (ap + bq)'(1) = ap'(l) +bq'(1) = a (0) + b(0) = 0 p, q in S rArr ap +bq in S then S is subspace q(x) = x - 2x2 + x3 q(1) = 1 - 2 + 1 = 0 q'(x)=1 - 4x + 3x2 q'(l) = 1 - 4 + 3 = 0 q is element of S

Need help with the double integration on problems (g,j,k,n,o,q, and x) thanks

be parametric esjuations for a tb) C i v i) C intersects esactly one e ine containing the point -1.3. intersects the xy and ntion voctor to find another esqua ne line with the Korm 17) Let C be the line the ine eontaining the point a) Give parametric ejuastions for c (b) Write an equation for C in on vector to find another equa for the same 18. Let C be the line deterimined by the tt (a) Provide a vector b) Give parametricequations fore c 19. Let C be the line determined by the the points where the line + bt.to t et) eplanes. Ia-0, esplain (a) Provide a vector (b) Write an equation for C in s, determine wbether the the distance between the c. 20. We wish to find the distance from C as shown in the figure that fo dinates of points P and Pa, but we do dinates of point Q (a) Ifyuknew the measure of angle Î¸ llows Wek sect, find the point of intent know th- t.5-3,6+7. weould find the distance from point P to lanes r2()3-,4+3t,5-2 Applications ne. Explain why the vation in Washington State. She tonic wastes in one of the tank tank farm is huge, and she d is leaking. Worse yet, the tank g the tanks would req tank farm, an operation that Instead, Emmy has wells dug the tank farm, to try to trace the earth's surface is conside the bottom of the tank farm the distance is given in f taminated groundwater at (1033, 247,-55). If the w of one of the tanks and i what is the location of th the line does not ow you can distance from point P to line C even if you do ne know the measure of angle Î¸. line containing use the tech- ization for the 24. P023.5), ds(23.5) 25,Pi-1, 3, 7. d = (2,0,4) the leak? eyuivalent.6. Ptbc 53. lan is climbing Mountl He has made a secon mountain. He does n 27, P3, 1), d= (2 5) 28, p(1.3,-2, 4), d= (4,-1,5, 8) In Exercises 29-34, find an equation of the line containing the Proofs ntaning the given pair of points. Express your answer (a) as a vector p ess your an- rametrization, (b) in terms of parametric equations,and ( n 54. Let Ci and C be line symmetric form. 29. P(O,0,0), Q(4,-1,6) and P2 and Qa poin C2 if and only if P â¢P(3,-1,7), Q(5,8,-2)