Objectives
- To understand how the unit normal vector of a vector valued function of one variable is defined.
- To identify the unit normal vector and its geometric properties relating to a curve in space.
Section 5.7 The Unit Normal Vector
This lesson focuses on measuring the direction of turning using the unit normal vector. The definition of the unit normal vector as the unit vector in the direction of change for the unit tangent is described and several examples are shown to demonstrate that acceleration is different than the unit normal. There are multiple choice questions that ask the user to think about when the unit normal vector is undefined and whether the unit normal vector is a property of the driver or the road (in our continuing analogy). The unit normal and unit tangent are animated on all four of our common examples.
Subsection 5.7.1 Unit Normal and Unit Tangent Vector Activities
Activity 5.7.1. Labeling T and N on 2D curves.
In this activity you will be looking at the curve and labels given in Figure 5.7.2. All of the points are labeled so they are traveled in alphabetical order. The points B and E are the same location achieved at different times on the curve.
(a)
For each of the five labeled points, draw and label \(\vec{T}\) and \(\vec{N}\text{.}\)
(b)
Put an “X” on any points where the unit normal vector will not exist and explain why the unit normal vector does not exist at that point.
(c)
Did you need to know how the curve in Figure 5.7.2 was parameterized to draw \(\vec{T}\) and \(\vec{N}\text{?}\) Does that mean that \(\vec{T}\) and \(\vec{N}\) are properties of the driver or road? Explain your answers.
