Lesson 2.2: Lines and Curves in Space

In this lesson, you will study vector-valued functions, space curves, and lines in space.

View all of the instructional videos, read the provided readings, and complete the practice exercises/activities. These will help you master the objectives for this lesson. Pay attention to additional comments provided.

Lesson Objectives

Upon completion of lesson 2.2, you will be able to:

• Determine equations of lines and line segments

• Analyze curves in space

• Determine intersections

• Match functions with graphs

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this lesson.

1. Introduction to Vector-Valued Functions [9:44]

   1. Source: bullcleo1 from YouTube

1. Position Vector Valued Functions [7:45]

   1. Source: KhanAcademy from YouTube

• 1. Warning: The presentation is sloppy and at times the language used is imprecise.

1. The Domain of a Vector Valued Function [5:35]

   1. Source: bullcleo1 from YouTube

1. Parametric Equations of a Line in 3D [8:13]

   1. Source: bullcleo1 from YouTube

• • • 1. Note: The orientation of the line is the direction in which the line is traced out as the parameter t increases. If you care about the orientation of a line, then you need to be careful with your choice of the direction vector v⃗ . If you do not care about the orientation of the line, then you may choose any nonzero vector that is parallel to the line as the direction vector v⃗ .

1. Finding the Vector Equation of a Line [7:41]

   1. Source: patrickJMT from YouTube

• 1. Note: The presenter made some minor computational errors, but he noted it on the video.

1. Finding the Point at Which a Line Intersects a Plane [5:38]

   1. Source: patrickJMT from YouTube

1. Vectors: Conditions for lines to be parallel [10:54]

   1. Source: ExamSolutions from YouTube

• 1. Note: Two lines are parallel if their direction vectors v1→ and v2→ are parallel, that is, if there exists a scalar k≠0 such that v1→=kv2→. In the video, the presenter mentioned "fixed ratio" which is the same idea.

1. Vectors: Intersecting and Skew Lines [12:48]

   1. Source: ExamSolutions from YouTube

• 1. Note: If two lines are neither parallel nor intersecting, then they are skew. In other words, the two lines do not lie in the same plane.

1. Vectors: Angle between two lines given their equations [5:27]

   1. Source: ExamSolutions from YouTube

Lesson Readings

The following required readings cover the content for this lesson. As you go through each reading, pay close attention to the content that will help you learn the objectives for this lesson.

1. Equations of Lines

   1. Paul's Online Math Notes

2. Vector-Valued Functions

• 1. This document covers materials from lessons in both Module 2 and Module 3.

• • Read slides 3-14 for this lesson.

  • Read slides 15-42 for Lesson 2.3.

  • Read slides 43- 66 for Lesson 3.1.

  • Read slides 90-102 for Lesson 3.2.

  • Read slides 67-89, 103-127 for Lesson 3.3.

Lesson Practice Exercises/Activities

Chapter 11: Vector –Valued Functions

Arc length will be covered in Lesson 3.2. Skip the part about the arc length in section 11.1 for now, practice exercise problems for section 11.1.

Additional Resources

Below are additional resources that help reinforce the content for this module.

1. Section 14.1 Vector Functions and Space Curves

   1. The limit part (from Definition 1.3 to Example 1.5) will be covered in Lesson 2.3.

2. Vector Equation of a line: Example 1 (video) [4:48]

• 1. Source: ExamSolutions

• 1. Math 254: Vector-Valued Equations

  2. A nice summary sheet for Module 2 and Module 3.

  3. Position Vector Valued Functions (video) [7:45]

  4. Source: Khan Academy

• 1. Vector Valued Functions (Read the Definition section of a Vector Valued Function only) After the first example click on the link "click here" to review some examples of parametric equations.

     1. Limit and continuity will be covered in the next lesson, Lesson 2.3.

        1. This document covers materials from lessons in both Module 2 and Module 3.

        2. Read slides 3-14 for this lesson.

        3. Read slides 15-42 for Lesson 2.3.

        4. Read slides 43- 66 for Lesson 3.1

        5. Read slides 90-102 for Lesson 3.2

        6. Read slides 67-89, 103-127 for Lesson 3.3

Homework Assignment

Do Homework 2.2 on MyMathLab.

Show Your Work Assignment

Show Your Work assignment will be given via Assignment Tool on Laulima. Please check your email for Laulima announcement.

Lesson 2.3: Calculus of Vector-Valued Functions -->