   Chapter 11.5, Problem 32E

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 − 2 − 3 = y − 2 6 = z − 3 , x − 3 2 = y + 5 = z + 2 4

To determine

To calculate: For the lines x23=y26=z3, x32=y+5=z+24 find the point of intersection of lines if they intersect and the angle between them.

Explanation

Given:

The parametric equations of the first line are:

x23=y26=z3

The parametric equations of the second line are:

x32=y+5=z+24

Formula used:

The angle between two lines is given by

cosθ=v1v2v1v2

Calculation:

Convert the symmetric equations of both the lines into parametric equations

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

Equations of the first line will be,

x23=y26=z3=t

x=23t,y=2+6t,z=t+3

And, the equation of the second line will be

x32=y+5=z+24=s

x=2s+3,y=s5,z=4s2

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

Now since, the lines intersect to each other at a point

The intersect point will be common for both the lines.

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

For x coordinate,

23t=2s+3 …… (1)

For y coordinate,

2+6t=s5 …… (2)

For z coordinate,

t+3=4s2 …… (3)

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

Then,

Form the equation (3),

t+3=4s2t=4s5

Now, the value of s can be determined by putting t=4s5 in equation (1) then,

23(4s5)=2s+3212s+15=2s+314s=14s=1

Therefore, the value of s is 1.

Now, with the value of s, then find the value of t,

Since,

t=4s5,

Then substitute the value of s and then find the value of t. Then,

t=4s5=4(1)5=45=1

Now, substitute t=1 and s=1 in the equation of first line and second line respectively. Then,

The x coordinate of for the first line is,

x=23t=2+3=5

The y coordinate of the first line is,

y=2+6t=26=4

The z coordinate of the first line is,

z=t+3=1+3=2

So, from the above calculation the coordinates for the first line are (5,4,2)

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