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Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

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BuyFindarrow_forward

Elementary Geometry For College St...

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

Explain why the lines below are coincident.

l 1 : x ,   y ,   z = 0 ,   0 ,   0 + n 1 ,   2 ,   - 3 and

l 2 : x ,   y ,   z = 1 ,   2 ,   - 3 + r - 1 ,   - 2 ,   3

To determine

To explain:

The reason for the given lines to be coincident.

Explanation

The given two lines are,

𝓁1: x, y, z=0, 0, 0+n1, 2, -3 and

𝓁2: x, y, z=1, 2, -3+r-1, -2, 3

Definition for coincident lines.

Two lines are said to be coincident if they have a common point and their direction vectors are 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=0+n, 0+2n, 0-3n

It can be written as,

𝓁1=n, 2n, -3n

𝓁2=1-r, 2-2r, -3+3r

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

n=1-r

2n=2-2r

-3n=-3+3r

From first equation, r=1-n

Substitute this value in second equation.

2n=2-21-n

Multiplying,

2n=2-2-2n

Combining the like terms,

2n+2n=2-2

Simplifying,

4n=0

From this,

n=0

Substitute n=0 in r=1-n.

r=1-0

Simplifying,

r=1

Substitute n=0 and r=1 in the point form of the line equations

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