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Math Help - Using Homogeneous Equations to prove the problems ?

  1. #1
    Kid
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    Using Homogeneous Equations to prove the problems ?

    Hi everyone,

    This time I have a little bit confuse when I try to solve this question that I think it is obviously shown:

    (a) Show that there is a line through any pair of points in the plane. [Hint: Every line has equation ax + by + c = 0, where a, b, c are not all zero].

    (b) Generalize and show that there is a plane ax + by + cz + d = 0 through any three points in space.

    (Source: Question 6, Section 1.3, Chapter 1, Linear Algebra with Applications 5th Edition, W. Keith Nicholson)

    At first I think it is so easy but after that I can not think how can I prove when it is too obviously. It made me going crazy and frustrated. Or maybe I have some problem in thinking (stupid or something ?) that I can not solve this.

    Thank you so much.
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  2. #2
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    When something is "too obvious", do the obvious! Let the two points be (x_0, y_0) and (x_1, y_1). If that line has equation ax+ by+ c= 0, then the equation must satisfy ax_0+ by_0+ c= 0 and ax_1+ by_1+ c= 0.

    Those are the same as ax_0+ by_0= -c and ax_1+ by_1= -c. As long as you can solve those equations for a and b, the line exists (and is unique). Under what conditions can you not solve those equations? Treat that case separately.
    Last edited by HallsofIvy; July 25th 2010 at 09:34 AM.
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  3. #3
    Kid
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    Thank you for you very logical and smart way.

    At first I think I must solve those equations but after that I asked myself "so what next ".

    Now I have an answer. Thank you so much!
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