Consider the plane through the points A = (0,1,1), B = (1,1,0), and C = (1,0,3), as well as lines along the plane in the directions of \(\overrightarrow{AB} \)and \(\overrightarrow{AC}\), spaced at integer multiples of those vectors:
The vectors v and w in the picture above are parallel to the plane. What are they equal to?
b). Find the value of x such that \(\mathbf{v} = \begin{pmatrix} 1 \\ 2 \\ x \end{pmatrix}\)is a vector parallel to the plane through the points A = (0,1,1), B = (1,1,0) and $C = (1,0,3).
c). Find the normal vectorn \(\mathbf{n} = \begin{pmatrix} n_1 \\n_2 \\ n_3 \end{pmatrix}\) to the plane through points A = (0,1,1), B = (1,1,0) and C = (1,0,3), such that \(n_1 + n_2 + n_3 = 5.\)
d). The equation of the plane through the points A = (0,1,1), B = (1,1,0) and C = (1,0,3) can be written as ax + by + cz = 8. Then what's the ordered triple (a, b, c)?
