Find equations of the following.3(x − z) = 12arctan(yz), (1 + π, 1, 1)(a) the tangent plane (b) parametric equations of the normal line to the given surface at the specified point. (enter your answer as a comma-separated list of equations. let x, y, and z be in terms of t.)

A.

B.

Step-by-step explanation:

A. At first, it´s useful to move everything to one side and name it as a function f(x,y,z):

:

To proceed to find the tangent plane at (1+π,1,1), we use the following equation for the tangent plane:

Where (x₀,y₀,z₀) is the specified point where we want the tangent plane to connect. Now we need to find the gradient vector of f:

Now we differentiate f with respect to x,y and z to find those coordinates:

We are ready to use the equation for the tangent plane

The tangent plane has an equation , and the orthogonal vector to this plane is one made of the coefficients of the plane, a normal vector for this plane is (-1,2,1).

To find a normal line to this surface in (1+π,1,1) we find a normal line to the plane, and because we know that (-1,2,1) is a normal vector, then the line has to have the same direction, so we normalize that vector to get the direction:

And because that line has to pass through (1+π,1,1) we conclude the vector equation for this line is the following:

and from this equation:

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step-by-step