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.
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.
+128474 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
