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Find the normal vector \(\mathbf{n} = \begin{pmatrix} n_1 \\n_2 \\ n_3 \end{pmatrix}\) to the plane through points A = (0,1,1), B = (1,1,0) and C = (1,0,3) such that n_1 + n_2 + n_3 = 5.

 Dec 24, 2019
 #1
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The cross product of A + B = (1,2,1) and B + C = (2,1,3) is (5,-1,-3).  We can then scale this to (25,-5,-15), to get the sum of the entries to be 5.

 Dec 24, 2019
 #2
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I tried that but got it wrong

Guest Dec 25, 2019
 #3
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BA =  (1, 0, -1)

CA  = ( 1, -1, 2)

 

Find the  cross-product   BA x CA

 

i     j      k        i         j

1    0     -1      1       0

1   -1      2      1       -1

 

[ 0i  - 1j  - 1k ]  - [ 0k + 1i + 2j ]  =   -1i - 3j - 1k   =   ( -1, -3 , -1)  = the  normal vector

 

 

Note that if we multiply this  vector by -1  we get

 

( 1, 3 , 1)

 

And adding these components gives us  5

 

 

cool cool cool

 Dec 25, 2019
edited by CPhill  Dec 25, 2019

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