By Gao J.

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A) Find the volume of the parallelepiped determined by the three vectors: a = (8324,5789,2098), b = (9265, -246, 8034), e = (4321, -765,7903). (b) Now consider all parallelepipeds whose base is determined by the vectors a = (2,0, -1) and b = (0,2, -1), and whose height is variable e = (x, y, z). Assume that x, y, and z are positive and lIeli = 1. Use the triple product to Problem Set B: Vectors and Graphics 31 compute a formula for the volume of the parallelepiped involving x, y, and z. Compute the maximum value of that volume in terms of x and y as follows.

24 Chapter 2: Vectors and Graphics 1n[29]:= ParametricPlot3D [Evaluate [total] , (t, 0, Pi}, Boxed -) False, Ticks -) None] ; That still isn't very good; however, a few minor changes will improve it significantly. The main problem is the viewpoint from which Mathematica has chosen to show us the graph. 4,2), where the units are multiples of the length of each displayed axis. This viewpoint is chosen generically, to make it unlikely that significant features of a random graph will be obscured. It has the definite disadvantage, however, that it changes the apparent directions of the :1;- and y-axes when compared with most mathematical textbooks.

At one point in the computation, we must divide by p', which explains why we assume p' f; O. ) Helical Curves Consider the right circular helix from our first example. It is clear that at every point on the helix, the unit tangent vector makes a constant angle with respect to the z-axis. With that in mind, we make the definition: a curve r is called a cylindrical helix if there is a fixed unit vector u such that T(s) . u is constant. Then we have Theorem 3. A curve with nonvanishing curvature is a cylindrical helix if and only if the ratio T / ~ is constant.

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