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Chapter 11: Problem 2
How many dependent scalar variables does the function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) have?
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Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\langle 2 \cos t, 2 \sin 3 t, 4 \cos 8 t\rangle$$
Use vectors to show that the midpoint of the line segment joining \(P\left(x_{1}, y_{1}\right)\) and \(Q\left(x_{2}, y_{2}\right)\) is the point \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\) (Hint: Let \(O\) be the origin and let \(M\) be the midpoint of \(P Q\). Draw a picture and show that $$\left.\overrightarrow{O M}=\overrightarrow{O P}+\frac{1}{2} \overrightarrow{P Q}=\overrightarrow{O P}+\frac{1}{2}(\overrightarrow{O Q}-\overrightarrow{O P}) \cdot\right)$$
Hexagonal circle packing The German mathematician Gauss proved that the densest way to pack circles with the same radius in the plane is to place the centers of the circles on a hexagonal grid (see figure). Some molecular structures use this packing or its three-dimensional analog. Assume all circles have a radius of 1 and let \(\mathbf{r}_{i j}\) be the vector that extends from the center of circle \(i\) to the center of circle \(j,\) for \(i, j=0,1, \ldots, 6\) a. Find \(\mathbf{r}_{0 j},\) for \(j=1,2, \ldots, 6\) b. Find \(\mathbf{r}_{12}, \mathbf{r}_{34},\) and \(\mathbf{r}_{61}\) c. Imagine circle 7 is added to the arrangement as shown in the figure. Find \(\mathbf{r}_{07}, \mathbf{r}_{17}, \mathbf{r}_{47},\) and \(\mathbf{r}_{75}\)
Jack pulls east on a rope attached to a camel with a force of 40 ib. Jill pulls north on a rope attached to the same camel with a force of 30 Ib. What is the magnitude and direction of the force on the camel? Assume the vectors lie in a horizontal plane.
Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle 5 t^{-4}-t^{2}, t^{6}-4 t^{3}, 2 / t\right\rangle$$
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