Find the unit vector normal to given surface f(x,y,z) = 16 at point (-1,-1,1).
Understand the Problem
The question is asking to find the unit vector normal to the surface defined by the function f(x, y, z) = x^2 + y^2 + z^2 at the specific point (-1, -1, 1). This involves calculating the gradient of the function at that point and normalizing it to get the unit vector.
Answer
The unit normal vector is: $$ -\frac{1}{\sqrt{3}} \mathbf{i} -\frac{1}{\sqrt{3}} \mathbf{j} + \frac{1}{\sqrt{3}} \mathbf{k} $$
Answer for screen readers
The unit vector normal to the surface at the point (-1, -1, 1) is: $$ \mathbf{n} = -\frac{1}{\sqrt{3}} \mathbf{i} - \frac{1}{\sqrt{3}} \mathbf{j} + \frac{1}{\sqrt{3}} \mathbf{k} $$
Steps to Solve
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Find the gradient of the function To find the unit normal vector, we first need to calculate the gradient of the function $f(x, y, z) = x^2 + y^2 + z^2$. The gradient is given by: $$ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) $$ Calculating the partial derivatives: $$ \frac{\partial f}{\partial x} = 2x $$ $$ \frac{\partial f}{\partial y} = 2y $$ $$ \frac{\partial f}{\partial z} = 2z $$
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Evaluate the gradient at the point (-1, -1, 1) Now, we evaluate the gradient at the point (-1, -1, 1): $$ \nabla f(-1, -1, 1) = \left( 2(-1), 2(-1), 2(1) \right) = (-2, -2, 2) $$
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Normalize the gradient vector to find the unit normal vector The next step is to normalize the gradient vector $(-2, -2, 2)$. The magnitude of the vector is: $$ ||\nabla f|| = \sqrt{(-2)^2 + (-2)^2 + (2)^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3} $$ The unit normal vector $\mathbf{n}$ is given by: $$ \mathbf{n} = \frac{\nabla f}{||\nabla f||} = \frac{(-2, -2, 2)}{2\sqrt{3}} = \left( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) $$
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Express the unit normal vector in vector notation The final unit normal vector can be written as: $$ \mathbf{n} = -\frac{1}{\sqrt{3}} \mathbf{i} - \frac{1}{\sqrt{3}} \mathbf{j} + \frac{1}{\sqrt{3}} \mathbf{k} $$
The unit vector normal to the surface at the point (-1, -1, 1) is: $$ \mathbf{n} = -\frac{1}{\sqrt{3}} \mathbf{i} - \frac{1}{\sqrt{3}} \mathbf{j} + \frac{1}{\sqrt{3}} \mathbf{k} $$
More Information
The process of finding the unit normal vector involves calculating the gradient of a surface and normalizing it. The normal vector is perpendicular to the surface and provides important information in vector calculus and physics.
Tips
- Not calculating the gradient correctly: Ensure all partial derivatives are calculated based on the function.
- Failing to normalize the vector: Remember to divide by the magnitude to get the unit vector.
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