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February 4th, 2018, 05:37 PM  #1 
Newbie Joined: Feb 2018 From: Canada Posts: 1 Thanks: 0  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(x1)2(y+2)+4(z3) And the general form of that would be 12x122y4+4z12 Which becomes 12x2y+4z28 = 0 Simplified to 6xy+2z14 = 0 However this answer is WRONG. The actual answer is 6x+y+2z10 = 0 The mistake I made is that the point normal form should be 12(x1)+2(y+2)+4(z3) 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 01:58 AM. 
February 5th, 2018, 01:39 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,298 Thanks: 1221 
I admit I don't really follow what you've done. The way I would attack this is form $v_1 = p_2p_1,~v_2 = p_3p_1$ $v_1 = (0,2,1),~v_2 = (2,6,3)$ $n = v_1 \times v_2 = (12,2,4)$ $n$ 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 $12(x1)+2(y+2) + 4(z3)=0$ or $12x + 2y + 4z = 20$ which reduces to $6x + y + 2z = 10$ 
February 5th, 2018, 07:01 AM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 555 Thanks: 319 Math Focus: Dynamical systems, analytic function theory, numerics 
Your 2by2 determinant computation is correct as you noticed but your 3by3 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 3by3 matrix. Read here for a reminder on how to compute determinants. https://en.wikipedia.org/wiki/Determinant 

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determinant, negative, positive 
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