Final answer:
To find the equation of the tangent plane to a surface S at the point (2,1,3) but don't have the equation, you would employ the normal vector. The correct option is B.
Step-by-step explanation:
To find the equation of the tangent plane to a surface S at the point (2,1,3) without having the equation, you would need to employ the normal vector. The equation of the tangent plane can be found by using the point-normal form, where the equation is given by (x-x0) n1 + (y-y0) n2 + (z-z0) n3 = 0, where (x0, y0, z0) is the point of tangency and (n1, n2, n3) is the normal vector to the surface at that point.
1. Find the normal vector to the surface at the given point (2,1,3). This can be done by taking the partial derivatives of the surface equation with respect to x, y, and z and evaluating them at (2,1,3).
2. Use the normal vector obtained in step 1 to write the equation of the tangent plane using the point-normal form. Plug in the coordinates of the point of tangency (2,1,3) and the components of the normal vector in the equation.
Therefore, to find the equation of the tangent plane to the surface S at the point (2,1,3), you would employ the normal vector.