Chapter 11: Q. 7 (page 879)

Given a differentiable vector-valued function $r (t)$ , what is the relationship between $r' (t_{0})$ and $T (t_{0})$ at a point $t_{0}$ in the domain of $r (t)$ ?

Short Answer

Expert verified

The relationship between $r' (t_{0}), T (t_{0})$ is $T (t) = \frac{r' (t)}{||r' (t)||}$ .

Step by step solution

Step 1. Given information.

Consider the given question,

A differentiable vector-valued function is $r (t)$ .

Step 2. Write the relationship between r't0,Tt0.

Assume $t_{0}$ be any point in the domain of $r (t)$ . The unit tangent vector of $r (t)$ is denoted by $T (t)$ .

According to the definition of unit tangent vector function on some interval $I \subseteq R$ such that $r' (t) \neq 0$ on I.

Then the unit tangent function can be defined as,

$T (t) = \frac{r' (t)}{||r' (t)||}$

Where, $r' (t_{0}) \neq 0$

Step 3. Consider the definition.

Consider the definition it is clear that,

Both $T (t_{0}), r' (t_{0})$ have same direction.

Also, role="math" localid="1649564983175" $r' (t_{0}), T (t_{0})$ may differ in lengths unless $||r' (t_{0})|| = 1$ , as $T (t_{0})$ always have the magnitude of $1$ unit because it is a unit vector of $r (t)$ .

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Given a differentiable vector-valued function $r (t)$ , what is the relationship between $r' (t_{0})$ and $T (t_{0})$ at a point $t_{0}$ in the domain of $r (t)$ ?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Write the relationship between r't0,Tt0.

Step 3. Consider the definition.

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Given a differentiable vector-valued function r(t), what is the relationship between r't0and Tt0at a pointt0in the domain of r(t)?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Write the relationship between r't0,Tt0.

Step 3. Consider the definition.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Given a differentiable vector-valued function $r (t)$ , what is the relationship between $r' (t_{0})$ and $T (t_{0})$ at a point $t_{0}$ in the domain of $r (t)$ ?