   Chapter 16.6, Problem 35E

Chapter
Section
Textbook Problem

Find an equation of the tangent plane to the given parametric surface at the specified point.35. r(u, v) = u cos v i + u sin v j + v k; u = 1, v = π/3

To determine

To find: An equation of the tangent plane to the vector function r(u,v)=ucosvi+usinvj+vk;u=1,v=π3 .

Explanation

Given data:

The vector function is given as follows.

r(u,v)=ucosvi+usinvj+vk;u=1,v=π3

Formula used:

Write the expression to find tangent plane to the parametric surface with the normal vector n=a,b,c at the specified point (x0,y0,z0) .

a(xx0)+b(yy0)+c(zz0)=0 (1)

Write the expression to find normal vector from the tangent vectors of the parametric surface.

n=|ijka1b1c1a2b2c2| (2)

Here,

The vector a1,b1,c1 is a tangent vector ru of the parametric surface and

The vector a2,b2,c2 is a tangent vector rv of the parametric surface.

Write the expression to find the tangent vector ru of the parametric surface.

ru=xui+yuj+zuk (3)

Write the expression to find the tangent vector rv of the parametric surface.

rv=xvi+yvj+zvk (4)

Write the vector function as follows.

r(u,v)=ucosvi+usinvj+vk;u=1,v=π3

Write the parametric equation from the vector function as follows.

x=ucosv,y=usinv,z=v

Calculation point in the surface (x0,y0,z0) :

Write the expression to find point (x0,y0,z0) in the surface.

(x0,y0,z0)=(ucosv,usinv,v)

Substitute 1 for u and π3 for v ,

(x0,y0,z0)=((1)cos(π3),(1)sin(π3),π3)=(12,32,π3)

Calculation of tangent vector ru :

Substitute ucosv for x , usinv for y , and v for z in equation (3),

ru=(ucosv)ui+(usinv)uj+(v)uk=[u(ucosv)]i+[u(usinv)]j+[u(v)]k=(cosv)i+(sinv)j+(0)k=cosv,sinv,0

Substitute 1 for u and π3 for v ,

ru=cos(π3),sin(π3),0=12,32,0

Calculation of tangent vector rv :

Substitute

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