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