Solve the problem.
-Find parametric equations for the normal line to the surface -6x - 7y - 6z = -11 at the point (1, -1, 2).
A) x = t - 6, y = -t - 7, z = 2t - 6
B) x = -6t + 1, y = -7t - 1, z = -6t + 2
C) x = -6t - 1, y = -7t + 1, z = -6t - 2
D) x = -t - 6, y = t - 7, z = -2t - 6

Question

Quizplus · Accepted Answer

First, we need to find the gradient vector of the surface at the given point, which is the same as the normal vector. We can do this by finding the partial derivatives of the surface equation:
-6x - 7y - 6z = -11
-6x/dx = 6
-7y/dy = 7
-6z/dz = 6
So the gradient vector at (1,-1,2) is <6,7,6>.
Next, we can use this vector as the direction vector for the normal line. Since we want the line to pass through the point (1,-1,2), we can use that as the initial point. So the parametric equations of the line are:
x = 1 + 6t
y = -1 + 7t
z = 2 + 6t
which matches choice B.

Solve the Problem. -Find Parametric Equations for the Normal Line to the Surface