Answer to Question #225910 in Chemical Engineering for Lokika

Question #225910

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)


1
Expert's answer
2021-08-17T12:30:01-0400

x²+y²+z²=3f(x,y,z)=x²+y²+z²3=0x² + y² + z² = 3\\ f ( x , y , z )= x² + y² + z² - 3=0

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

The gradient is obtained as follows

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

=2,2,22+2+2=28,28,28=12,12,12=\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}}


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