Final answer:
To write the equation of the tangent plane to a parametrized surface, calculate the partial derivatives of the surface function, take their cross product to find the normal vector, and use this with the point of tangency to define the plane.
Step-by-step explanation:
To find the equation of the tangent plane to a parametrized surface at a specific point, you must first know the parametric equations that define the surface. If, for example, the surface is given by a vector function R(u, v), where u and v are parameters, you need to determine the partial derivatives of R with respect to u and v at the point of interest. These partial derivatives are vector-valued functions that indicate the direction of the surface tangent in their respective parameter dimensions. Taking the cross product of these vectors gives the normal vector to the surface at that point. The tangent plane is defined by the point on the surface and the normal vector. The equation of the tangent plane can be expressed as (r - r0) · n = 0, where r is a position vector to any point on the tangent plane, r0 is the position vector to the given point on the surface, and n is the normal vector obtained from the cross product.