Question

Show the complete solution on the given problems.

1. Find the equation of the tangent plane and the parametric equations of the normal line to

Answer 1

Let's say our surface is $z=3x^2-xy$ and we want to find tangent plane and the parametric equations of the normal line in (1,2,1). So $F(x,y,z)=3x^2-xy-z$ .

1) Gradient of F is (6x-y, -x, -1), grad F(1,2,1) = (4,-1,-1). So tangent plane will have the equation that is a scalar multiplication of grad F(1,2,1) on (x-1,y-2,z-1) because (1,2,1) is a point in which we're looking for tangent plane, so 4(x-1)-(y-2)-(z-1)=0, which is 4x-y-z-1=0. Answer is 4x-y-z=1.

2) Each of the equations x(t), y(t), z(t) will be the sum of x, y, z component of a point (1,2,1) and corresponding of vector grad F(1,2,1) multiplied by parameter t. So if grad F(1,2,1)=(4,-1,-1) and point (1,2,1) then the parametric equations of the normal line will look like this:

$x(t)=1+4t, \newline y(t)=2-t, \newline z(t)=1-t.$