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Chapter 12: Problem 3

Given a tangent vector on an oriented curve, how do you find the unit tangent vector?

Short Answer

Expert verified
Answer: The steps to find the unit tangent vector of an oriented curve at a specific point are: 1. Determine the function representing the curve, r(t). 2. Find the derivative of the function, r'(t). 3. Evaluate the derivative at the point of interest, r'(t_0). 4. Calculate the magnitude of the tangent vector, ||r'(t_0)||. 5. Normalize the tangent vector to find the unit tangent vector, T(t_0) = r'(t_0) / ||r'(t_0)||.

Step by step solution

01

Determine the function representing the curve

We need to know the function representing the curve whose unit tangent vector we want to find. Let's denote this function as r(t), where t is a parameter, and r(t) provides the position of points on the curve.
02

Find the derivative of the function

To find the tangent vector, we need to calculate the first derivative of r(t) with respect to the parameter t. Let's denote this derivative as r'(t).
03

Evaluate the derivative at the point of interest

If we want to find the tangent vector at a specific point on the curve, we need to evaluate r'(t) at that point. So let's say we are interested in the tangent vector at t = t_0, then we need to find r'(t_0).
04

Calculate the magnitude of the tangent vector

Before we can normalize the tangent vector to have a length of 1, we need to determine its current magnitude, or length. The magnitude of the tangent vector, ||r'(t_0)||, can be calculated using the formula: ||r'(t_0)|| = \sqrt{(r'_x(t_0))^2 + (r'_y(t_0))^2 + (r'_z(t_0))^2} where r'_x(t_0), r'_y(t_0), and r'_z(t_0) are the x, y, and z components of r'(t_0) respectively.
05

Normalize the tangent vector to find the unit tangent vector

To normalize the tangent vector and find the unit tangent vector, divide each component of r'(t_0) by its magnitude: T(t_0) = \frac{r'(t_0)}{||r'(t_0)||} The resulting vector, T(t_0), is the unit tangent vector at the specified point on the curve.

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Most popular questions from this chapter

\(\mathbb{R}^{2}\) Consider the vectors \(\mathbf{I}=\langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle\) and \(\mathbf{J}=\langle-1 / \sqrt{2}, 1 / \sqrt{2}\rangle\). Write the vector \langle 2,-6\rangle in terms of \(\mathbf{I}\) and \(\mathbf{J}\).

Zero curvature Prove that the curve $$ \mathbf{r}(t)=\left\langle a+b t^{p}, c+d t^{p}, e+f t^{p}\right\rangle $$ where \(a, b, c, d, e,\) and \(f\) are real numbers and \(p\) is a positive integer, has zero curvature. Give an explanation.

Parabolic trajectory Consider the parabolic trajectory $$ x=\left(V_{0} \cos \alpha\right) t, y=\left(V_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2} $$ where \(V_{0}\) is the initial speed, \(\alpha\) is the angle of launch, and \(g\) is the acceleration due to gravity. Consider all times \([0, T]\) for which \(y \geq 0\) a. Find and graph the speed, for \(0 \leq t \leq T.\) b. Find and graph the curvature, for \(0 \leq t \leq T.\) c. At what times (if any) do the speed and curvature have maximum and minimum values?

In contrast to the proof in Exercise \(81,\) we now use coordinates and position vectors to prove the same result. Without loss of generality, let \(P\left(x_{1}, y_{1}, 0\right)\) and \(Q\left(x_{2}, y_{2}, 0\right)\) be two points in the \(x y\) -plane and let \(R\left(x_{3}, y_{3}, z_{3}\right)\) be a third point, such that \(P, Q,\) and \(R\) do not lie on a line. Consider \(\triangle P Q R\). a. Let \(M_{1}\) be the midpoint of the side \(P Q\). Find the coordinates of \(M_{1}\) and the components of the vector \(\overrightarrow{R M}_{1}\) b. Find the vector \(\overrightarrow{O Z}_{1}\) from the origin to the point \(Z_{1}\) two-thirds of the way along \(\overrightarrow{R M}_{1}\). c. Repeat the calculation of part (b) with the midpoint \(M_{2}\) of \(R Q\) and the vector \(\overrightarrow{P M}_{2}\) to obtain the vector \(\overrightarrow{O Z}_{2}\) d. Repeat the calculation of part (b) with the midpoint \(M_{3}\) of \(P R\) and the vector \(\overline{Q M}_{3}\) to obtain the vector \(\overrightarrow{O Z}_{3}\) e. Conclude that the medians of \(\triangle P Q R\) intersect at a point. Give the coordinates of the point. f. With \(P(2,4,0), Q(4,1,0),\) and \(R(6,3,4),\) find the point at which the medians of \(\triangle P Q R\) intersect.

Find the points (if they exist) at which the following planes and curves intersect. $$y+x=0 ; \mathbf{r}(t)=\langle\cos t, \sin t, t\rangle, \text { for } 0 \leq t \leq 4 \pi$$
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