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If we want to specify the equation of a plane that goes through 3 different points we have the following determinant: $$\begin{vmatrix}x-a1&y-a2&z-a3\\b1-a1&b2-a2&b3-a3\\c1-a1&c2-a2&c3-a3 \end{vmatrix}=0$$ which comes from the mixed product of the vectors. But we can also use the following determinant: $$\begin{vmatrix}x&y&z&1\\a1&a2&a3&1\\b1&b2&b3&1\\c1&c2&c3&1 \end{vmatrix}=0$$ How can we go from the first equation to the second one?

Gianniskido
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  • This is a formula that comes from projective geometry: the 1s in the last column are the 4th coordinates in projective $3$-space for points at finite distance. – Bernard Jan 15 '18 at 19:00
  • So there in no way of transforming the first equation to the second? Is it just another approach? – Gianniskido Jan 15 '18 at 19:04
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    You can; just subtract the second row from the first, third and fourth rows, and expand the determinant by the last column. – Bernard Jan 15 '18 at 19:09

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