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# Linear Algebra I 1016 331

RIT

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This 2 page Class Notes was uploaded by Mr. Eladio Murphy on Monday October 19, 2015. The Class Notes belongs to 1016 331 at Rochester Institute of Technology taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/225063/1016-331-rochester-institute-of-technology in Applied Math And Statistics at Rochester Institute of Technology.

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Date Created: 10/19/15

Matrix Algebra 1016 331 Fall 2005 Instructor David S Hart Geometry Handout We ve seen that various curves and surfaces can be given by a determinant equation For example the X line through 1 2 and 3 4 is given by 1 0 What follows is a summary of what we ve 3 21 41 done along with further items you might want to explore A conic curve in the plane has an equation of the form 2 2 ax bxycy dxeyf0 2 The express1on b 4a c tells us what type of con1c 1t 1s In part1cular three outcomes are g1ven by gt 0 hyperbola b2 4ac 0 parabola lt 0 ellipse This must be used with care and only after first making sure that we actually have a conic and not something else The equation xy 1 satisfies b2 4ac 0 yet it represents a line not a parabola A similar remark applies to x2 2xyy2 1 Do you see what f1gure satisfies this equation It is not a parabola even though b2 4ac 0 Lastly notice that x2 y2 1 has no solutions in the real numbers The fact that b2 4 ac 4 lt 0 means nothing and the equation certainly does not represent an ellipse Identify the curves x2xyy2 3x4 x22xyy2 3x4 I x23xyy2 3x4 In class I showed that it takes ve points to determine a conic Unless the conic has special properties fewer points are not enough Confirm this for yourself with the three curves just given Find four points shared by all of them So for each of these curves a fifth point on it will be the tie breaker Pick ve points all with integer coordinates on each of the above curves and write down the determinant form of the equation Choose your points carefully making good use of zeros Evaluate the detemlinants you get Remember that we can trade information for points If we know something about a curve then we need fewer points to detennine its equation A parabola that opens up or down has its central axis parallel to the y axis and its directrix parallel to the x axis Perhaps you recall some features of the equation of this type of parabola If so you will see at once that b 0 and c 0 Therefore just three points are needed to find its equation Write down the determinant form of the equation of such a parabola that passes through the points 110 1 4 and 314 Evaluate the determinant to get the usual form of the equation The equation of a quadric surface in space has the form ax2bxycxzdy2eyzf22gxhyizj0 So nine points correctly chosen will determine a quadric surface The choice of points must not be made without care Two different spheres for example may have infinitely many points in common If so then no number of points chosen from this common stock will distinguish them Still when wisely chosen nine points suffice Just as before knowing what restrictions we have lessens the number of points we need If it is a sphere we want then a d f and b c e 0 So a sphere can be determined by four points Do you see why the four points must not all lie on a circle If they do then four points will not be enough to determine a sphere In fact no number of such points all on a circle would be enough Consider the two surfaces x2 y2 222 25 an ellipsoid x2 y2 222 25 a hyperboloid of one sheet Find eight points all with integer coordinates that are shared by these two surfaces On each surface find a ninth point again with integer coordinates that does not lie on the other Use these points or some modification of them to give the determinant equation of each surface Evaluate the determinants Are you surprised by the results In this problem you may need several attempts to get the equations you expect Try varying your selections of nine points in a thoughtful way You may want to use a computer algebra system like Maple to help evaluate the determinants you get If you enjoy problems like these here is a last one Use a determinant approach to find a simple equation of the quadric surface that passes through the following nine points Identify the surface x0011 1 122 2 y 1 1 0 2 0 2 1 1 1 z 1 10 6 2 4 1 1 3

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