Cramers Rule and Determinants - A more detailed analysis

[succubus]

I know I am not presenting an actual problem, but it is for homework, and I do need some help. I wasn't sure which forum to post in, so I posted in two. :( Sorry.

I am doing a presentation on the 3-point problem in Geology. We have to use Cramers Rule to solve for the equation of a plane. I can do it no problem, but since I have to present it, I want to be prepared to answer all questions my teacher may ask. For example:

Why is the equation of a plane written like this

| x y z 1|
|x1 y1 z1 1|
|x2 y2 z2 1|
|x3 y3 z3 1|

The top is just the vector we multiply by to come up with the equation of a line? (when we expand by cofactors) I'm not sure exactly. Unfortunately I did not take the required pre-req to get into this class, they let me in, so I took on a project that was a little beyond me at this point. Also, what is the definition of a determinant? I can show how they work, ubt I don't have a description of what they really represent. Everything I look up and every book I have just show you how to analyze them, not what they really represent.

Any help on the explanation of this would be great.


-Pati

 

[HallsofIvy]

First, of course, what you have is NOT the equation of a plane because it is not even an equation. It is, rather, one of the determinants you would use, in Cramer's method, to solve several equations.

The equation of any (almost) plane can be written Ax+ By+ Cz= 1 for some choice of A, B, and C. If the plane includes the 3 points [itex](x_1,y_1,z_1)[/itex], [itex](x_2,y_2,z_2)[/itex], and [itex](x_3,y_3,z_3)[/itex] then you must have [itex]Ax_1+ By_1+ Cz_1= 1[/itex], [itex]Ax_2+ By_2+ Cz_2= 1[/itex], and [itex]Ax_3+ By_3+ Cz_3= 1[/itex]. Including the generic Ax+ By+ Cz= 1 gives 4 equations whose augmented matrix is what you have above.

You are correct that expanding the determinant by minors, using the first row gives the determinants that appear in Cramer's rule. It is, in fact, how Cramer's rule is derived.

 

[succubus]

Aahh...well, I meant why is the equation of a plane found using a determinant such as that. ;) I'm a bit on edge as I have to present this soon, and I have been racking my brain. Thanks for your input though, the more input themerrier!