Question

A unit vector which is perpendicular to the surface of the paraboloid of revolution $z = x^2 + y^2$ at the point (1,2,5) is

• A. $\pm \frac{1}{\sqrt{21}} (2\hat{i} + 4\hat{j} - \hat{k})$
• B. $\pm \frac{1}{\sqrt{21}} (4\hat{i} + 2\hat{j} - \hat{k})$
• C. $\pm \frac{1}{\sqrt{21}} (2\hat{i} + 2\hat{j} - \hat{k})$
• D. $\pm \frac{1}{\sqrt{21}} (2\hat{i} + 4\hat{j} + \hat{k})$

Hint:

To find a vector normal to any surface $z = g(x,y)$, rewrite it as $f(x,y,z) = g(x,y) - z = 0$.
The gradient $\nabla f$ will always have a $-1$ coefficient for the $\hat{k}$ term.
This allows you to quickly eliminate incorrect options.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find a unit vector that is normal (perpendicular) to the surface of the paraboloid $z = x^2 + y^2$ at the specific point $P(1, 2, 5)$.

Step 2: Key Formula or Approach:
For any level surface represented by $f(x, y, z) = C$, the gradient vector $\nabla f$ at a point $P$ is perpendicular to the surface at that point.
The unit normal vector $\hat{n}$ is obtained by dividing the gradient vector by its magnitude:
\[ \hat{n} = \pm \frac{\nabla f}{|\nabla f|} \]

Step 3: Detailed Explanation:

• Define the function $f(x, y, z)$ by rearranging the surface equation:
\[ f(x, y, z) = x^2 + y^2 - z = 0 \]
• Find the gradient vector $\nabla f$ by taking partial derivatives:
\[ \nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k} \] \[ \nabla f = 2x\hat{i} + 2y\hat{j} - \hat{k} \]
• Evaluate this gradient vector at the given point $P(1, 2, 5)$:
\[ \nabla f|_{(1,2,5)} = 2(1)\hat{i} + 2(2)\hat{j} - \hat{k} = 2\hat{i} + 4\hat{j} - \hat{k} \]
• Calculate the magnitude of the gradient vector:
\[ |\nabla f| = \sqrt{2^2 + 4^2 + (-1)^2} = \sqrt{4 + 16 + 1} = \sqrt{21} \]
• Write down the unit normal vector $\hat{n}$:
\[ \hat{n} = \pm \frac{2\hat{i} + 4\hat{j} - \hat{k}}{\sqrt{21}} \]

Step 4: Final Answer:
The unit normal vector perpendicular to the surface is $\pm \frac{1}{\sqrt{21}} (2\hat{i} + 4\hat{j} - \hat{k})$.