Answer to Question #106528 in Calculus for Joseph Se

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:

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