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Math Help - Help to prove

  1. #1
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    Help to prove

    I have to prove if a,b,c <> 0 and
    \frac{{ay - bx}}{c} = \frac{{cx - az}}{b} = \frac{{bz - cy}}{a} then is \frac{x}{a} = \frac{y}{b} = \frac{z}{c}

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  2. #2
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    Write the common value \frac{{ay - bx}}{c} = \frac{{cx - az}}{b} = \frac{{bz - cy}}{a} = t and treat it as a system of three linear equations in four unknowns x,y,z,t. Note that the rank of the system is three, since the determinant of the 3-by-3 minor corresponding to x,y,z is 2abc: hence there is a one-dimensional family of solutions. Since x=a, y=b, z=c, t=0 is clearly a solution, all solutions are multiples of this.
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  3. #3
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    Quote Originally Posted by rgep
    Write the common value \frac{{ay - bx}}{c} = \frac{{cx - az}}{b} = \frac{{bz - cy}}{a} = t and treat it as a system of three linear equations in four unknowns x,y,z,t. Note that the rank of the system is three, since the determinant of the 3-by-3 minor corresponding to x,y,z is 2abc: hence there is a one-dimensional family of solutions. Since x=a, y=b, z=c, t=0 is clearly a solution, all solutions are multiples of this.
    Can you explain me solution little bit more?
    I didn't quite understand it.
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