   Chapter 11, Problem 10PS

Chapter
Section
Textbook Problem

# Using Parametric Equations Consider the line given by the parametric equations x = − t + 3 , y = 1 2 t + 1 , z = 2 t − 1 and the point (4, 3, s) for any real number s.(a) Write the distance between the point and the line as a function of s.(b) Use a graphing utility to graph the function in part (a). Use the graph to find the value of s such that the distance between the point and the line is minimum.(c) Use the zoom feature of a graphing utility to zoom out several times on the graph in part (b). Does it appear that the graph has slant asymptotes? Explain. If it appears to have slant asymptotes, find them.

(a)

To determine

To calculate: The distance between the line as a function of s and the given point (4,3,s) such that x=t+3,y=12t,z=2t1.

Explanation

Given:

The parametric equations form given equation,

x=t+3,y=12t,z=2t1

Also,

The point (4, 3, s) for any real number s.

Formula used:

The distance is found by D=PQ¯×uu

Calculation:

Consider the parametric equations form given equation,

x=t+3,y=12t+1,z=2t1

Now use the direction numbers which is 1,12,2.

Hence, the direction vector for the line is,

u=(1,12,2)=(2,1,4)

Now, Point on the line;

Let,

t=0

Then,

P=(3,1,1)

Therefore,

PQ=(43,31,s(1))=(1,2,s+1)

The cross product is,

(b)

To determine

To graph: The function D=5s2+10s+11021 also use the graph to calculate the value of s such that the distance between the point and the line is minimum.

(c)

To determine

To calculate: The slant asymptotesfrom the graph of part (b).

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