Question

Find a unit vector normal to the surface of the ellipsoid at (2,2,1) if the ellipsoid is defined as f(x,y,z) = x^2 + y^2 + z^2 – 10.

(a) \(\frac{2}{3} a_x + \frac{2}{3} a_y + \frac{1}{3} a_z \)

(b) \(\frac{1}{3} a_x + \frac{1}{3} a_y + \frac{1}{3} a_z \)

(c) \(\frac{2}{3} a_x + \frac{2}{3} a_y + \frac{2}{3} a_z \)

(d) \(\frac{2}{3} a_x + \frac{1}{3} a_y + \frac{1}{3} a_z \)

This question was posed to me in an interview for job.

This intriguing question comes from Gradient of a Function and Conservative Field topic in chapter Vector Differential Calculus of Engineering Mathematics

Answer 1

Right option is (a) \(\frac{2}{3} a_x + \frac{2}{3} a_y + \frac{1}{3} a_z \)

For explanation: First we have to find the gradient of the function, which is equal to 2x ax + 2y ay + 2z az.

Gradient of f at (2,2,1) is (4,4,2).

\(a_n= \frac{±(4,4,2)}{6} =  \frac{2}{3} a_x + \frac{2}{3} a_y + \frac{1}{3} a_z. \)