text: Q:
  
  Equations of lines in 3 dimensions with vectors
  
  The intersection of the planes defined by
  $$v• \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$$
  and
  $$v• \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$$
  is a line. Find an equation of this line.
  I have taken the dot products like so
  $$\begin{pmatrix} x \\ y \\ z \end{pmatrix}• \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 8x+1y-12z = 35$$
  $$\begin{pmatrix} x \\ y \\ z \end{pmatrix}• \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 6x+y-9z=70$$
  But now have no idea on where to go! I thought about using systems of equations, but there are three variables and two equations! Any help would be appreciated!
  
  A:
  
  Your equation set is given by
  \begin{align}
  8v_{1} + v_{2} - 12v_{3} &= 35 \\
  6v_{1} + 7v_{2} - 9v_{3} &= 70
  \end{align}
  As you stated, you have less equations than variables, so we set one of the variables to be a parameter. Setting $z = t$, we get
  \begin{align}
  8v_{1} + v_{2} &= 35 + 12t \\
  6v_{1} + 7v_{2} &= 70 + 9t
  \end{align}
  Solving this last system yields
  \begin{align}
  v_{1} &= \frac{7 + 3t}{2} \\
  v_{2} &= 7
  \end{align}
  Hence the equation of the line is given by
  \begin{align}
  \vec{v} &= \left( \frac{7 + 3t}{2}, 7, t \right) \\
  &= \left( \frac{7}{2}, 7, 0 \right) + t \left( \frac{3}{2}, 0, 1 \right)
  \end{align}
  
  