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Chapter 12: Problem 4

Interpret the principal unit normal vector of a curve. Is it a scalar function or a vector function?

Short Answer

Expert verified
Explain your answer. Answer: The principal unit normal vector is a vector function. This is because its output is a vector that depends on the input of the curve's points and the corresponding tangent vectors, providing information about the direction of the curvature of the curve.

Step by step solution

01

Define the principal unit normal vector

The principal unit normal vector is a vector that is perpendicular to the tangent vector of a curve at a specific point, having a magnitude (length) of 1. It provides information about the direction of the curvature of the curve.
02

Explain the properties of the principal unit normal vector

The principal unit normal vector, denoted as \(\boldsymbol{N}\), can be obtained by normalizing the derivative of the unit tangent vector, \(\boldsymbol{T}\), with respect to the arc length parameter, \(s\). The formula for the principal unit normal vector is given by: \(\boldsymbol{N} = \frac{d\boldsymbol{T}}{ds}\div \left\lVert \frac{d\boldsymbol{T}}{ds}\right\rVert \) Here, \(\frac{d\boldsymbol{T}}{ds}\) represents the rate of change of the tangent vector with respect to the arc length, and the denominator normalizes the vector to have a magnitude of 1.
03

Determine if the principal unit normal vector is a scalar function or a vector function

A scalar function takes one or more input variables and produces a single scalar (real number) output, while a vector function takes one or more input variables and produces a vector output (a set of scalars). With the given definition and formula for the principal unit normal vector, we can see that the output of the function is a vector (\(\boldsymbol{N}\)) that depends on the input of the curve's points and the corresponding tangent vectors. Therefore, the principal unit normal vector is a vector function, not a scalar function.

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