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February 4th, 2018, 04:37 PM   #1
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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



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.

Last edited by skipjack; February 5th, 2018 at 12:58 AM.
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February 5th, 2018, 12:39 AM   #2
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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

$6x + y + 2z = 10$
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February 5th, 2018, 01:42 AM   #3
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Quote:
Originally Posted by MontanaMax View Post
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.
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February 5th, 2018, 06:01 AM   #4
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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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