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-   -   Negative determinant is positive? (http://mymathforum.com/linear-algebra/343347-negative-determinant-positive.html)

MontanaMax February 4th, 2018 04:37 PM

Negative determinant is positive?
 
Hello, I am looking at this question.

"Determine a general form of the equation of the plane through each of the following sets of three points"

P1(1,-2,3)
P2(1,0,2)
P3(-1,4,6)

This seems simple enough. I get this matrix

https://i.imgur.com/rehV9zn.png

Figuring out the determinants for each will get me
12,-2,4

So putting them in point normal form SHOULD net me

12(x-1)-2(y+2)+4(z-3)

And the general form of that would be

12x-12-2y-4+4z-12
Which becomes

12x-2y+4z-28 = 0
Simplified to
6x-y+2z-14 = 0

However this answer is WRONG. The actual answer is
6x+y+2z-10 = 0

The mistake I made is that the point normal form should be
12(x-1)+2(y+2)+4(z-3)

But I don't understand why. The determinant of the matrix
0 -1
-2 3
Is unquestionably -2. So why is the point normal form using +2 instead? I am especially confused because I have done other questions like this and have gotten answer correct without having to switch around signs. So what is happening here that I'm not seeing? Help would be much appreciated, thank you.

romsek February 5th, 2018 12:39 AM

I admit I don't really follow what you've done.

The way I would attack this is form

$v_1 = p_2-p_1,~v_2 = p_3-p_1$

$v_1 = (0,2,-1),~v_2 = (-2,6,3)$

$n = v_1 \times v_2 = (12,2,4)$

$n$ is the normal vector to the plane

Thus, using $p_1$ as a point on the plane, the scalar equation of the plane is given by

$n\cdot ((x,y,z)-p_1) = 0$

or

$12(x-1)+2(y+2) + 4(z-3)=0$

or

$12x + 2y + 4z = 20$

which reduces to

$6x + y + 2z = 10$

skipjack February 5th, 2018 01:42 AM

Quote:

Originally Posted by MontanaMax (Post 588119)
So why is the point normal form using +2 instead?

You should always alternate the signs. That's how determinant evaluation is done.

For the benefit of others, your method is briefly described here.

SDK February 5th, 2018 06:01 AM

Your 2-by-2 determinant computation is correct as you noticed but your 3-by-3 determinant is where you are making the mistake. Namely, as you compute minors for the matrix they should alternate in sign. The second minor is -2 as you noticed but should gain a minus sign from the determinant computation of the 3-by-3 matrix.

Read here for a reminder on how to compute determinants.
https://en.wikipedia.org/wiki/Determinant


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