Question

The unit normal to the surface x² + y² + z² = 3 at (1,1,1) is A)1/√16 (2i+2j+2k) B)1/√3(i+j+ k)

C) 1/√6(i+2j+3k)

D)1/√3(i+2j+2k)

Answer 1

$x² + y² + z² = 3\\ f ( x , y , z )= x² + y² + z² - 3=0$

The gradient of $f ( x , y , z )$  at point $x , y , z$  is a vector normal to the surface at this point.

The gradient is obtained as follows

$∇ f ( x , y , z ) = ( f_ x , f_ y , f_ z ) = 2x+2y+2z$ at point (1,1,1) and the unit vector is

$=\frac{2,2,2}{\sqrt{2+2+2}}\\ =\frac{2}{\sqrt{8}}, \frac{2}{\sqrt{8}},\frac{2}{\sqrt{8}}\\ =\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$