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Chapter 14: Q 18. (page 1106)

Give a smooth parametrizationr(t) for the straight line between the points蟺,e,1and(3,7,13).

Short Answer

Expert verified

The required parametrization is r(t)=鉄�蟺+(3-蟺)t,e+(7-e)t,1+12t鉄�,0鈮�t鈮�1

Step by step solution

01

Given Information

Smooth Parametrizationr(t)for straight line between the points(蟺,e,1)and(3,7,13).

02

Write parametric form of the equation of a line segment with two points

The parametric form of the equation of a line segment with two points x1,y1,z1and x2,y2,z2is

x=(1-t)x1+tx2,y=(1-t)y1+ty2,z=(1-t)z1+tz2

Now,

x1,y1,z1=(蟺,e,1)and x2,y2,z2=(3,7,13)

鈬�x=(1-t)x1+tx2

=(1-t)(蟺)+t(3)

=蟺-蟺t+3t

=蟺+(3-蟺)t

03

Complete the Parametrization

Also,

y=(1-t)y1+ty2

=(1-t)(e)+t(7)

=e-et+7t

=e+(7-e)t

And

z=(1-t)z1+tz2

=(1-t)(1)+t(13)

=1-t+13t

=1+12t

The parametric form is

x=蟺+(3-蟺)t,y=e+(7-e)t,z=1+12t

Hence,

r(t)=鉄�蟺+(3-蟺)t,e+(7-e)t,1+12t鉄�,0鈮�t鈮�1

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