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Chapter 11: Problem 4

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

Short Answer

Expert verified
Answer: The principal unit normal vector is a vector function.

Step by step solution

01

Definition of Principal Unit Normal Vector

The principal unit normal vector is a unit vector, orthogonal (at right angles) to the tangent vector, that describes the curvature of the curve at a specific point. It's defined as the derivative of the unit tangent vector divided by the magnitude of the derivative of the unit tangent vector.
02

Properties of Principal Unit Normal Vector

Some of the properties of the principal unit normal vector are: 1. It is orthogonal to the tangent vector at the point where it is defined. 2. The magnitude of the principal unit normal vector is always 1 (because of the name "UNIT" normal vector). 3. The principal unit normal vector points towards the center of curvature and changes direction as we move through the curve.
03

Scalar or Vector Function

A scalar function assigns a scalar (a real number) to each point in a domain, whereas a vector function assigns a vector to each point in a domain. In the case of the principal unit normal vector, at each point on the curve, we have a vector assigned to it which points in the direction of the center of curvature with a magnitude of 1. Therefore, the principal unit normal vector is a VECTOR FUNCTION.

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Most popular questions from this chapter

Use the formula in Exercise 79 to find the (least) distance between the given point \(Q\) and line \(\mathbf{r}\). $$Q(-5,2,9) ; \mathbf{r}(t)=\langle 5 t+7,2-t, 12 t+4\rangle$$

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