   Chapter 10.6, Problem 32E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Explain why the lines below are skew. l 1 : x ,   y ,   z = - 2 ,   - 2 ,   - 2 + n 1 ,   2 ,   3 and l 2 : x ,   y ,   z = 1 ,   1 ,   1 + r 1 ,   - 3 ,   5

To determine

To explain:

The reason for the given lines to be skew.

Explanation

The given two lines are,

𝓁1: x, y, z=-2, -2, -2+n1, 2, 3 and

𝓁2: x, y, z=1, 1, 1+r1, -3, 5

Definition for skew lines.

Two lines are said to be skew if they have no common point and their direction vectors are not multiples of each other.

We have to check whether there is any common point occurs for the two lines.

Write the point form of the two lines.

𝓁1=-2+n, -2+2n, -2+3n

𝓁2=1+r, 1-3r, 1+5r

If there is a common point for the two lines, then

-2+n=1+r

-2+2n=1-3r

-2+3n=1+5r

From first equation, r=-2+n-1

r=-3+n

Substitute this value in second equation.

-2+2n=1-3-3+n

Multiplying,

-2+2n=1+9-3n

Combining the like terms,

2n+3n=1+9+2

5n=12

n=125

From this,

n=125

Substitute n=125 in r=-3+n

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