   Chapter 11.5, Problem 31E

Chapter
Section
Textbook Problem

# Finding a Point of IntersectionIn Exercises 33–36, determine whether the lines intersect, and if so, find the point of intersection and the angle between the lines. x 3 = y − 2 − 1 = z + 1 , x − 1 4 = y + 2 = z + 3 − 3

To determine

To calculate: For the lines x3=y21=z+1 find the point of intersection of lines if they intersect and the angle between them.

Explanation

Given:

The symmetric equations of the first line are:

x3=y21=z+1

The symmetric equations of the second line are:

x14=y+2=z+33

Formula used:

The angle between two lines is given by

cosθ=v1v2v1v2

Calculation:

Let us first convert the symmetric equations of both the lines into parametric equations

This can be attained by equating the equations with t and s respectively.

Equation of first line will be,

x3=y21=z+1=t

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

And, equation of second line will be

x14=y+2=z+33=s

x=4s+1,y=s2,z=3s3

For the intersect point will be common for both the lines,

The lines should intersect to each other at a point

Then, the coordinates x,y and z for both the lines will be equal at the point of intersection.

Now, equate x,y and z coordinates of both the lines. Then,

For x coordinate,

3t=4s+1 …… (1)

For y coordinate,

2t=s2 …… (2)

For z coordinate,

t1=3s3 …… (3)

Since, by y coordinate, the relation between t and s can be easily determined.

Then,

Form the equation (2),

2t=s24=t+st=4s

Now, the value of s can be determined by putting t=4s in equation (3). Then,

t1=3s34s1=3s33+3=2ss=3

Therefore, the value of s is 3

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