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Chapter 12: Problem 3
Compute \langle 2,3,-6\rangle\(\cdot\langle 1,-8,3\rangle\).
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Determine whether the following statements are true and give an explanation or counterexample. a. The line \(\mathbf{r}(t)=\langle 3,-1,4\rangle+t\langle 6,-2,8\rangle\) passes through the origin. b. Any two nonparallel lines in \(\mathbb{R}^{3}\) intersect. c. The curve \(\mathbf{r}(t)=\left\langle e^{-t}, \sin t,-\cos t\right\rangle\) approaches a circle as \(t \rightarrow \infty\). d. If \(\mathbf{r}(t)=e^{-t^{2}}\langle 1,1,1\rangle\) then \(\lim _{t \rightarrow \infty} \mathbf{r}(t)=\lim _{t \rightarrow-\infty} \mathbf{r}(t)\).
For the given points \(P, Q,\) and \(R,\) find the approximate measurements of the angles of \(\triangle P Q R\). $$P(1,-4), Q(2,7), R(-2,2)$$
For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle-1,2,3\rangle, \mathbf{v}=\langle 2,1,1\rangle\)
Consider the curve described by the vector function \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k},\) for \(t \geq 0\). a. What is the initial point of the path corresponding to \(\mathbf{r}(0) ?\) b. What is \(\lim _{t \rightarrow \infty} \mathbf{r}(t) ?\) c. Sketch the curve. d. Eliminate the parameter \(t\) to show that \(z=5-r / 10\), where \(r^{2}=x^{2}+y^{2}\).
An object moves clockwise around a circle centered at the origin with radius \(5 \mathrm{m}\) beginning at the point (0,5) a. Find a position function \(\mathbf{r}\) that describes the motion if the object moves with a constant speed, completing 1 lap every 12 s. b. Find a position function \(\mathbf{r}\) that describes the motion if it occurs with speed \(e^{-t}\)
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