Math Problem Statement
Solution
Let's analyze the problem and solve it step by step.
Problem Summary:
You are asked to find:
- The equation of the tangent plane to the surface at the point .
- The equation of the normal line at the same point.
Step 1: Tangent Plane Equation
The general form of the tangent plane to a surface defined implicitly as at a point is given by: Where:
- are partial derivatives of with respect to , respectively.
For the given surface, we can rewrite the equation as: This is because the surface is given explicitly for .
Partial Derivatives:
- (since is explicitly given)
Evaluate at :
Thus, the equation of the tangent plane at is: Simplifying: Now, divide through by the coefficient of (which is already 1) to satisfy the condition "make the coefficient of equal to 1":
Step 2: Normal Line Equation
The normal line passes through the point and has a direction vector equal to the gradient of the surface at that point. The gradient is the vector of partial derivatives: So the parametric equation of the normal line is: This expands to:
Final Answer:
- Tangent Plane Equation:
- Normal Line Equation:
Would you like more details on any of these steps? Here are some related questions you might want to explore:
- What is the geometric interpretation of the tangent plane and normal line?
- How are the partial derivatives used to find the normal vector?
- How does the gradient relate to the direction of the normal line?
- Why does the coefficient of need to be set to 1 for the tangent plane equation?
- What would the tangent plane and normal line look like graphically?
Tip: The gradient vector is always perpendicular (normal) to the surface at any given point, making it a crucial tool for both tangent planes and normal lines.
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Tangent Planes
Normal Lines
Partial Derivatives
Gradient Vectors
Formulas
Tangent Plane Equation: Fx(x0, y0, z0)(x - x0) + Fy(x0, y0, z0)(y - y0) + Fz(x0, y0, z0)(z - z0) = 0
Normal Line Equation: Parametric form r(t) = (x0, y0, z0) + t(gradF)
Theorems
Gradient Vector Theorem
Suitable Grade Level
College/University Level (Multivariable Calculus)
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