Question

If f=xyz then its normal at(1,2,3)is

Answer 1

Given:

f = xyz.

Point ( Xo , Yo, Zo ) = ( 1 ,2 , 3)

To Find:

Normal at ( 1, 2, 3)

Solution:

If F(x,y,z) is a surface that is differentiable at  point (Xo,Yo,Zo), then the normal to F(x,y,z) at ( x0 , y0 , z0 ) is the line with normal vector

•   GradientF(Xo,Yo,Zo)

passing through the point (Xo,Yo,Zo).

Therefore the equation of normal will be,

• x(t) = x0 + Fx(x0,y0,z0) t  

• y(t) = y0 + Fy(x0,y0,z0) t    
• z(t) = z0 + Fz(x0,y0,z0) t

Therefore,

GradientF = Partial derivative of F with respect to x , y  and z .

GradientF ( xyz at (1,2,3) ) = < yz , xz , xy > = < 6 , 3 , 2 >

Therefore,

• Fx = 6
• Fy = 3
• Fz = 2

The parametric form of the normal line is,

• x(t) = 1 + 6 t  

• y(t) = 2+ 3 t    

• z(t) = 3 + 2 t

Therefore the equation of normal line is $\frac{x - 1 }{6} = \frac{y - 2}{3} = \frac{z - 3}{2}$