Crames Rule and equation of a plane

[succubus]

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

 

[Chris Hillman]

Attempting to answer a technically tricky question in an intuitive way

I am doing a presentation on the 3-point problem in Geology.

Geometry? As in: find the equation of the plane through three given points in ordinary three-dimensional euclidean space?

Why is the equation of a plane [in $R^3$] written like this
$$\left| \begin{array}{cccc} x & y & z & 1 \\ x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 &1 \end{array} \right| = 0$$

Well, can you answer this? Why is the equation of the line through two points in the plane $R^2$ given by
$$\left| \begin{array}{ccc} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{array} \right| = 0$$

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.

Well, proving the key property of determinants for nxn matrices, the multiplication law
$$\det (A \, B) = \left( \det A \right) \, \left( \det B \right)$$
is a bit tricky. The preferred modern proof is nowadays done in a course on exterior algebra (Grassmann algebra). But there is a lovely and elementary proof from graph theory! (Can't find the citation right now, but I can't be the only one who is thinking of the paper I am thinking of, so speak up, please, if you think you might recall the citation I intend.) When I taught linear algebra, I explained this in terms of a "hierarchical counting problem", but this interpretation is more abstract than geometrical.

As for intuition: geometrically speaking, the determinant is kind of like a signed volume; see section 10.3 of Birkhoff and Mac Lane, A Survey of Modern Algebra, 4th ed., MacMillan, 1977 for a proof of the following:

Theorem (Gram's Formula). The squared volume of the parallipiped in $R^n$ with edges given by the rows of the nxn matrix A is $\det A A^t$

(This is a generalization of the cosine law from elementary trigonmetry.)

Theorem. The linear operator P multiplies the volumes of all parallipipeds by $\pm \det P$ (depending on whether or not P reverses orientation).

If you know about elementary matrices, as an exercise you can try to prove these yourself by considering more and more elaborate matrices.

 

[succubus]

Geometry? As in: find the equation of the plane through three given points in ordinary three-dimensional euclidean space?


The answer is yes. I guess I should have been more precise. You see, were doing the three point problem in Geology. You have 3 elevations and 3 coordinates (x,y) and you need to find strike, dip, and dip direction for the plane. Basically, we have to find the equation of the plane in order to determine those other elements. Which comes upon another question I have, which is regarding the strike. (Don't see clearly why you take the arctan of the partial derivative of x with respect to y instead of the opposite) In any case, I just need to be able to explain (at least I think) why the equation of a plane is written like that. And the equation of 2 points for that matter. So the answer is no, I don't see why it's written that way :)

 

[Chris Hillman]

Ow, ow, ow (revenge of the dead trees!) ... Mrmph... eh? Oh, sorry, I've been trying to find that citation and so far can't, arghghgh! Probably one of those nifty papers by Philip Straffin...

Anyway, from similar triangles, the equation of the line through the points
$$(x_1, \, y_1), \; (x_2, \, y_2)$$
is (draw a sketch!):
$$\frac{x-x_1}{x-x_2} = \frac{y-y_1}{y-y_2}$$
On the other hand
$$\left| \begin{array}{ccc} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{array} \right| = \left| \begin{array}{ccc} x & y & 1 \\ x-x_1 & y-y_1 & 0 \\ x-x_2 & y-y_2 & 0 \end{array} \right| = \left| \begin{array}{cc} x-x_1 & y-y_1 \\ x-x_2 & y-y_2 \end{array} \right|$$
which vanishes iff the equation of the line holds true. (This is Exercise 10.1.10 in Birkhoff and Mac Lane, incidentally.)

Here's a hint for the higher dimensional case: consider difference vectors wrt the undetermined point (x,y,z); what property do they have iff (x,y,z) lies in the plane spanning the three given points?

I should have asked whether you have a preference for row vectors or column vectors (in the latter case, transpose the matrices in the argument I wrote out above).

I am afraid I am not familiar with the terms from geology (surveying, perchance?) "dip", "strike".

For the hyperplane through n points in $R^n$ we put n difference vectors in an n by n determinant and rewrite the latter as an n+1 by n+1 determinant as above, then apply the remark about the difference vectors having a certain familiar algebraic property iff the undetermined point lies in the hyperplane. Statement and proof are really about convex combinations of vectors so this is a fact about n-dimensional affine geometry. No notion of angle or distance is needed!

For an algebraic interpretation of basically the same argument, see M. G. Kendall, A Course in the Geometry of n Dimensions, Dover reprint.