   Chapter 12.1, Problem 57E

Chapter
Section
Textbook Problem

# Representing a Graph by Vector-Valued Function In exercises 55-62 sketch the space curve represented by the intersection of the surfaces a. Then represent the curve by a vector valued function using the given parameter.Surfaces Parameter x 2 + y 2 + z 2 = 4 , x + z = 2 x = 1 + sin t

To determine

To Calculate: The space curve represented by the intersection of the surfaces and represent the curve by a vector valued function.

Explanation

Given: Surface is x2+y2+z2=4,x+z=2 and the parameter is x=1+sint.

Graph: Black coloured curve is the required curve of intersection (done by mathematica)

Calculation:

The intersection of curve is represented by,

r(t)=(1+sint)i+2costj+(1sint)k.

Le t x2+y2+z2=4...(1),x+z=2..(2)

Put x=1+sint in equation (2),

1+sint+z=2z=21sint=1sintz=1sint

Put the value of x and z in (1),

(1+sint)2+y2+(1sint)2=4

(1+sint

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