To determine the equations of a line that passes through M r (2.1,-1) and is orthogonal to the plan, ret, and r: and t:
Start with the most general form of a Plane
Ax + By + Cz + D = 0
Orthogonal to r
(1) 3A + 2B + C = 0
Orthogonal to t
6A + 4B + 2C = 0 ==> 3A + 2B + C = 0
Ah! They played a little trick on us. r and t are parallel. Fortunately, we can generate more direction numbers form the two most obvious points on the lines.
2-0 = 2, -1-0 = -1, and 1-0 = 1
Orthogonal to a line with those dirction numbers.
(2) 2A - B + C = 0
Contains the point (2,1,-1)
(3) 2A + B - C + D = 0
That's enough. With three constraints and four variables, we'll have to pick one to parameterize the others. I chose A.
B = -A/3, C = -7A/3, and D = -4A
Substituting into the original general form and dividing by -A/3 gives:
3x - y - 7z - 12 = 0
There are the values we seek. The line is then trivial.