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Chapter 10: Problem 18

Find the principal unit normal vector to the curve at the specified value of the parameter. $$ \mathbf{r}(t)=t \mathbf{i}+\frac{6}{t} \mathbf{j}, \quad t=3 $$

Short Answer

Expert verified
The principal unit normal vector to the curve for \(t=3\) can be determined by following the indicated steps, which involve derivative calculations, finding the magnitude of the derivative, and normalization of vectors.

Step by step solution

01

Find the derivative

The first step is to take the derivative of the vector function, \(\mathbf{r}(t)=t \mathbf{i}+\frac{6}{t} \mathbf{j}\). The derivative \(\mathbf{r}'(t)\) is obtained by calculating the derivative of each component of the vector function with respect to \(t\), to yield \(\mathbf{r}'(t) = \mathbf{i} - \frac{6}{t^2} \mathbf{j}\).
02

Find the magnitude of the derivative

Next, calculate the magnitude of \(\mathbf{r}'(t)\). Use the formula for the magnitude of a vector, \(||\mathbf{r}'(t)||=\sqrt{\left(\frac{d}{dt} \right)^2 + \left(-\frac{6}{t^2}\right)^2}\). Substituting \(t=3\) into the expression to get the magnitude, the calculation yields physically meaningful value.
03

Calculate the Unit Tangential Vector

The unit tangent vector \(T(t)\) is given by the formula \(\mathbf{T}(t)=\frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}\), where |\(\mathbf{r}'(t)|\) is the magnitude of \(\mathbf{r}'(t)\) calculated in the previous step. Evaluate \(\mathbf{T}(t)\) for \(t=3\).
04

Calculate the derivative of the Unit Tangential Vector

Now, compute the derivative of \(\mathbf{T}(t)\) to obtain \(\mathbf{T}'(t)\). Substituting \(t=3\) to evaluate \(\mathbf{T}'(3)\).
05

Calculate the Unit Normal Vector

The unit normal vector \(\mathbf{N}(t)\) is given by the formula \(\mathbf{N}(t)=\frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}\). Substitute the values found earlier and compute for the unit normal vector at \(t=3\).

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Most popular questions from this chapter

Use the given acceleration function to find the velocity and position vectors. Then find the position at time \(t=2\) $$ \begin{array}{l} \mathbf{a}(t)=t \mathbf{j}+t \mathbf{k} \\ \mathbf{v}(1)=5 \mathbf{j}, \quad \mathbf{r}(1)=\mathbf{0} \end{array} $$

Find the angle \(\theta\) between \(r(t)\) and \(r^{\prime}(t)\) as a function of \(t .\) Use a graphing utility to graph \(\theta(t) .\) Use the graph to find any extrema of the function. Find any values of \(t\) at which the vectors are orthogonal. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j} $$

Use the model for projectile motion, assuming there is no air resistance. \([a(t)=-9.8\) meters per second per second \(]\) A projectile is fired from ground level at an angle of \(8^{\circ}\) with the horizontal. The projectile is to have a range of 50 meters. Find the minimum velocity necessary.

A projectile is launched with an initial velocity of 100 feet per second at a height of 5 feet and at an angle of \(30^{\circ}\) with the horizontal. (a) Determine the vector-valued function for the path of the projectile. (b) Use a graphing utility to graph the path and approximate the maximum height and range of the projectile. (c) Find \(\mathbf{v}(t),\|\mathbf{v}(t)\|,\) and \(\mathbf{a}(t)\) (d) Use a graphing utility to complete the table. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{t} & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 \\ \hline \text { Speed } & & & & & & \\ \hline \end{array} $$ (e) Use a graphing utility to graph the scalar functions \(a_{\mathbf{T}}\) and \(a_{\mathrm{N}} .\) How is the speed of the projectile changing when \(a_{\mathrm{T}}\) and \(a_{\mathbf{N}}\) have opposite signs?

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=\frac{1}{t-1} \mathbf{i}+3 t \mathbf{j} $$
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