Final answer:
To find the value of 'a', we need the equation of the surface and the gradient vector to find the surface normal.
Step-by-step explanation:
In three-dimensional space, a vector is defined as the sum of its three components: the x-component along the x-axis, the y-component along the y-axis, and the z-component along the z-axis. In this case, the given vector v = 7i + 3j + ak is tangent to the surface S at point P (2,2) above. To determine the value of a, we need to find the z-component of the vector.
Since the vector is tangent to the surface at point P, the z-component should be parallel to the surface normal at that point. The surface normal vector can be obtained by finding the gradient of the surface equation. Once we have the surface normal vector, we can equate it to the given vector and solve for a to find its value.
Without the equation of the surface, it is not possible to determine the exact value of a.