31 x y z 1 1 t 1 1 1 and 3 2 1 32 x y z t 1 1 1 and 3

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31. (x, y, z) = (1,0,1) +t(1,1,1), and (3,2,1)32. (x, y, z) =t(1,1,1), and (3,2,1)33. (x, y, z) = (4,-1,5) +t(2,0,7), and (1,1,1)34. (x, y, z) = (4,-1,5) +t(1,2,3), and (4,-1,6)In each of the following exercises, consider a rigid rod, with one end at theorigin and the other end at(x1, y1,0), where the coordinates are measured inmeters. Suppose a force of F= (a, b,0)Newtons is applied at(x1, y1,0). Then,calculate the magnitude of the torque produced by F around the origin, andindicate whether the torque vector comes out the page towards you, or goesinto the page (where we take the page to be thexy-plane).35. (x1, y1) = (0,1), (a, b) = (3,4).36. (x1, y1) = (1,0), (a, b) = (3,4).37. (x1, y1) = (1,2), (a, b) = (3,4).38. (x1, y1) = (1,-2), (a, b) = (-4,3).More Depth:39. Find the volume of the parallelepiped spanned by the vectors (1,0,1), (2,-1,2),and (5,3,0).40. Find the volume of the parallelepiped spanned by the vectors (7,1,-2), (1,1,1),and (3,2,1).41. Find the volume of the parallelepiped with one vertex at (1,0,1) and adjoiningvertices at (7,1,4), (1,1,1), and (3,2,1).42. Find the volume of the parallelepiped with one vertex at (-1,1,2) and adjoiningvertices at (0,0,0), (1,1,4), and (3,2,5).43. Find the volume of the tetrahedron with vertices at (0,0,0), (1,0,0), (0,1,0) and(0,0,1).
88CHAPTER 1.MULTIVARIABLE SPACES AND FUNCTIONS44. Find the volume of the tetrahedron with vertices at (1,-1,0), (2,0,0), (0,2,1)and (3,4,0).+Linear Algebra:45. Find the 4-dimensional volume of the parallelotope spanned by the vectors (1,0,1,0),(0,2,-1,2), (0,5,3,0), and (1,2,3,4).46. Find the 5-dimensional volume of the parallelotope spanned by the vectors (1,0,1,0,1),(0,2,-1,2,0), (0,5,3,0,-1), (1,2,3,4,5), and (4,0,0,-2,1).
1.6.FUNCTIONS OF A SINGLE VARIABLE891.6Multi-Component Functions of a Single VariableMultivariable Calculus is the Calculus of functions of more than one variable.Thismeans that the domains of the functions should be subsets ofRn, wheren2.However, in this section, we look at functions where the domain of the function is asubset ofRand the codomain is a subset ofRn, wheren2. This reduces to the studyofnsingle-variable functions, and so is not truly a “multivariable” topic. You may haveencountered much of this material in your studies of single-variable Calculus.Afunctionf:A!Bis spec-ified by giving a domainA,a codomainB, and a rulefthat associates, to each ele-ment ofA, a unique elementofB.Thecodomainmaycontain more elements thantherangeoff.Basics:Suppose that a particle is moving in space in such a way that itsx,y, andzco-ordinates, measured in meters, at timetseconds (measured from some initial startingtime), are given byx=x(t) = cost,y=y(t) = sint,andz=z(t) =t.Then, the position of the particle is given by the functionp:R!R3given byp(t) = (x(t), y(t), z(t)) = (cost,sint, t).More generally, asingle-variable function intoRnis any functionp:A!B, whereAis a subset ofRandBis a subset ofRn. We also refer to functions intoRn, wheren2, asmulti-component functions.In this context, a function which takes valuesinR, instead of in a higher-dimensional Euclidean space, is referred to as areal-valuedfunction.

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