Chapter 10: Q .8. (page 833)

Explain how the direction vectors of two lines in $ℝ^{3}$ can be used to determine whether the two lines are parallel or identical. If the two lines are not parallel or identical, what are the possibilities for them?

Short Answer

Expert verified

If the direction vectors of the two lines are not multiples of each other, then the lines are either intersecting or skew lines.

Step by step solution

Step 1:Given information

The direction vectors of two lines in $ℝ^{3}$

Step 2:Explaination

In $3 d -$ space the lines either intersect, are parallel, or are identical. There is also one more possibility that the lines may be skew lines.

For any given two line equations we use the direction vectors to know the relation between then If the direction vectors of two lines are scalar multiples of each other, then the lines are either parallel or identical.

If the direction vectors of the two lines are not multiples of each other, then the lines are either intersecting or skew lines.

Therefore, if the two lines are not parallel or identical, then the possibility is they are intersecting or skew lines.

Hence this is the explanation.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Explain how the direction vectors of two lines in $ℝ^{3}$ can be used to determine whether the two lines are parallel or identical. If the two lines are not parallel or identical, what are the possibilities for them?

Short Answer

Step by step solution

Step 1:Given information

Step 2:Explaination

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Statistics

Calculus

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Explain how the direction vectors of two lines in ℝ3 can be used to determine whether the two lines are parallel or identical. If the two lines are not parallel or identical, what are the possibilities for them?

Short Answer

Step by step solution

Step 1:Given information

Step 2:Explaination

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Statistics

Calculus

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Explain how the direction vectors of two lines in $ℝ^{3}$ can be used to determine whether the two lines are parallel or identical. If the two lines are not parallel or identical, what are the possibilities for them?