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Course: Calculus III
Topic: Vectors and Geometry of Space
Subtopic: Lines and Curves

Overview

The wonders of three dimensional objects! Expanding lines and curves from 2D to 3D, describing and analyzing curves in 3D, studying special 3D curves such as the helix. Merging vectors, functions, and parametric equations into vector-valued function used to track points in space as they spin out curves. These are among the visually exciting things we get to explore.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 13.5.1 Find equation of the line through point P0 in direction of vector vecv in vecr=vec(r_o)+t vecv (vector) form
• 13.5.2 Find equation of the line through point P parallel to vector vecv in both parametric and symmetric forms
• 13.5.3 Find equation of the line through two points in both parametric and symmetric forms
• 13.5.4 Find equation of line segment
• 13.5.5 Understand and be able to describe the orientation of a curve (eg. positive = forward)
• 13.5.6 Be familiar with some special 3D curves and their equations including helix curve and slinky curve

Terminology

Define: orientation of a curve

Formulae to have in your notes: equation of a line (vector, parametric, and symmetric forms):

 Equation of a Line in 3D through point P_o (x_o,y_o,z_o) parallel to vector vecv=<> Vector form vecr=vec(r_o)+t vecv vecr=<>+t<> Parametric form x=x_o +at y=y_o +bt z=z_o +ct Symmetric form(x-x_o)/a=(y-y_o)/b=(z-z_o)/c

Supplemental Resources (recommended)

Download: Briggs Calc for STEM's Multivariable Study Card If you print this it is only page 1 and half of page 2 that is used in this course. The rest is used in Calc IV.

Supplemental Resources (optional)

Paul's OL Notes - Calc III: Equations of Lines

Patrick JMT Just Math Tutorials: Finding the Vector Equation of a Line

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down right column to "Vectors in Space". Starting about half-way through that list has some related titles. In particular Parametric Equations of Lines in 3D.