Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line (x + 3) /2 = (y ‒ 3) /2 = (z ‒ 2) /5?

Answer 1

The equation of a line is l(t)=r_0+tr, where the vector r is parallel to the line. This is found by taking the three terms you have for x,y,z and re-solving for x,y,z in terms of t e.g. (x+3)/2=t implies x=2t-3. It can be seen right for the equation that r=<2,2,5> (the numbers in the denominators). Then the vector between the two points is <3,3,1>.

In order for the the plane to be parallel to the line, the vector between the two points and the vector that the line is parallel to would also have to be parallel

Check <2,2,5>x<3,3,1>=<-13,13,0> not equal to zero

Since the vectors are not parallel, it isn't possible to have a plane that is parallel to the line. The line would intersect this plane.

Answer 2

Three points are needed to determine the equation of a plane.