Find the intersection of the plane and line given

Question

r= (0,3,7) + t(1,2,-1)

2x-6y-z-2=0

Please include steps and answers.

Answer 1

Because r = (0,3,7) + t(1,2,-1) then parametric equation of line: x = t, y = 3 + 2t, z = 7 - t. We substitude now these values in equation of plane to get value of parametr t corresponding to intersectionpoint:

2t - 6(3 + 2t) - (7 - t) - 2 = 0, 2t - 18 - 12t - 7 + t - 2 = 0; - 9t = 27, t = - 3.

So we can get coordinates of point of intersection: x = - 3, y = 3 + 2(-3) = -3, z = 7 - (- 3) = 10.

Answer: Point of intersection (- 3, - 3, 10)