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Paint USS128 In ObjectiveC Using Barcode creation for iPad Control to generate, create GS1 128 image in iPad applications. Print GS1  13 In C# Using Barcode generation for .NET framework Control to generate, create EAN13 image in .NET framework applications. and (y /3)2 = sin2 t When we add these two equations, lefttoleft and righttoright, we get (x /2)2 + (y /3)2 = cos2 t + sin2 t The rules of trigonometry tell us that cos2 t + sin2 t = 1 so the preceding equation can be rewritten as (x /2)2 + (y /3)2 = 1 and further morphed into x2/4 + y2/9 = 1 This equation describes an ellipse in the xy plane whose horizontal (xcoordinate) semiaxis measures 2 units, and whose vertical (ycoordinate) semiaxis measures 3 units Now let s consider the z coordinate The equation for z in terms of t is z = 3t This equation tells us that a point on our object travels in the negative z direction as the value of the parameter t increases The complete set of three parametric equations therefore describes an elliptical helix centered on the z axis As we move in the positive z direction, the helix rotates clockwise, because the coefficient of t is negative It looks something like the helix in Fig 1816, except that it s stretched by approximately 50 percent in the positive and negative y directions (vertically in this particular illustration) EAN128 Generator In Java Using Barcode creator for Java Control to generate, create UCC.EAN  128 image in Java applications. Painting Code 3/9 In None Using Barcode drawer for Office Excel Control to generate, create Code 39 image in Office Excel applications. Here s an extracredit challenge! Reading Data Matrix 2d Barcode In Visual Studio .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET framework applications. Encode UPCA In VS .NET Using Barcode printer for ASP.NET Control to generate, create UPC Code image in ASP.NET applications. Sketch threedimensional perspective graphs of the helixes described in the foregoing two examples and challenge Barcode Encoder In ObjectiveC Using Barcode generator for iPad Control to generate, create barcode image in iPad applications. Code 128 Code Set C Encoder In None Using Barcode maker for Online Control to generate, create Code 128C image in Online applications. Solution
You re on your own That s why you get extra credit! Lines and Curves in ThreeSpace
Practice Exercises
This is an openbook quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find workedout answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Consider the following threeway equation for a straight line in Cartesian xyz space: x 1=y 2=z 4 Find a point on the line, find the preferred direction numbers, and determine the direction vector as a sum of multiples of i, j, and k 2 Consider the following threeway equation for a straight line in Cartesian xyz space: 4x = 5y = 6z Find a point on the line, find the preferred direction numbers, and determine the direction vector as a sum of multiples of i, j, and k 3 Consider the following threeway equation for a straight line in Cartesian xyz space: (x 2)/3 = (4y 8)/4 = (z + 5)/( 2) Find a point on the line, find the preferred direction numbers, and determine the direction vector as a sum of multiples of i, j, and k 4 Consider a relation in Cartesian xyz space described by the system of parametric equations x = 4 y=t z = t2 1 Draw a twodimensional graph of this relation as it appears when we look broadside at the plane containing it 5 Consider a relation in Cartesian xyz space described by the system of parametric equations x = t2 + 2t y=t z=0

