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To solve this problem, we first need to determine the normal vector to the surface of the sphere at the point \( P(2, 2, 2) \). The equation of the sphere is:

\[ (x - 1)^2 + y^2 + z^2 = 9 \]

The gradient of the function \( f(x, y, z) = (x - 1)^2 + y^2 + z^2 - 9 \) gives us the normal vector to the surface of the sphere at any point. The gradient is calculated as follows:

\[ \nabla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \]

First, compute the partial derivatives:

\[ \frac{\partial f}{\partial x} = 2(x - 1) \]
\[ \frac{\partial f}{\partial y} = 2y \]
\[ \frac{\partial f}{\partial z} = 2z \]

Therefore, the gradient vector at any point \((x, y, z)\) is:

\[ \nabla f(x, y, z) = (2(x - 1), 2y, 2z) \]

At the point \( P(2, 2, 2) \):

\[ \nabla f(2, 2, 2) = (2(2 - 1), 2 \cdot 2, 2 \cdot 2) = (2, 4, 4) \]

The tangent line at point \( P(2, 2, 2) \) will be perpendicular to this normal vector, meaning the direction vector of the tangent line will be perpendicular to \( (2, 4, 4) \).

We need to check which of the given vectors are perpendicular to \( (2, 4, 4) \). Two vectors are perpendicular if their dot product is zero.

Calculate the dot product of each given vector with \( (2, 4, 4) \):

a) \( (4, 2, -4) \):

\[ (2, 4, 4) \cdot (4, 2, -4) = 2 \cdot 4 + 4 \cdot 2 + 4 \cdot (-4) = 8 + 8 - 16 = 0 \]

b) \( (4, -1, -1) \):

\[ (2, 4, 4) \cdot (4, -1, -1) = 2 \cdot 4 + 4 \cdot (-1) + 4 \cdot (-1) = 8 - 4 - 4 = 0 \]

c) \( (-8, 1, 3) \):

\[ (2, 4, 4) \cdot (-8, 1, 3) = 2 \cdot (-8) + 4 \cdot 1 + 4 \cdot 3 = -16 + 4 + 12 = 0 \]

d) \( (-1, 3, -3) \):

\[ (2, 4, 4) \cdot (-1, 3, -3) = 2 \cdot (-1) + 4 \cdot 3 + 4 \cdot (-3) = -2 + 12 - 12 = -2 \neq 0 \]

Therefore, the correct answer choices are:

a) \( (4, 2, -4) \)
b) \( (4, -1, -1) \)
c) \( (-8, 1, 3) \)

Would you like further details on this solution or have any additional questions?

Here are 5 more questions you might find interesting:
1. How do you find the normal vector of a plane?
2. What is the dot product of two vectors and how is it used?
3. Can you explain the gradient and its significance in calculus?
4. How do you find the equation of a tangent line to a sphere?
5. What are the properties of perpendicular vectors in three-dimensional space?

**Tip:** When dealing with vectors, remember that the dot product can help determine orthogonality (perpendicularity) by checking if the result is zero.

Learn how to find the tangent line to a sphere at a given point and identify perpendicular vectors using the dot product. This problem covers vector calculus, normal vectors, tangent lines, and perpendicular vectors in three-dimensional space. Suitable for college-level mathematics.

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