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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Evaluate the following definite integrals. $$\int_{0}^{\ln 2}\left(e^{t} \mathbf{i}+e^{t} \cos \left(\pi e^{t}\right) \mathbf{j}\right) d t$$
Vectors \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) for lines a. If \(\mathbf{r}(t)=\langle a t, b t, c t\rangle\) with \(\langle a, b, c\rangle \neq\langle 0,0,0\rangle,\) show that the angle between \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) is constant for all \(t>0\) b. If \(\mathbf{r}(t)=\left\langle x_{0}+a t, y_{0}+b t, z_{0}+c t\right\rangle,\) where \(x_{0}, y_{0},\) and \(z_{0}\) are not all zero, show that the angle between \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) varies with \(t\) c. Explain the results of parts (a) and (b) geometrically.
Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\langle 0,2,2 t\rangle ; \mathbf{r}(1)=\langle 4,3,-5\rangle$$
Suppose the vector-valued function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) is smooth on an interval containing the point \(t_{0} .\) The line tangent to \(\mathbf{r}(t)\) at \(t=t_{0}\) is the line parallel to the tangent vector \(\mathbf{r}^{\prime}\left(t_{0}\right)\) that passes through \(\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right) .\) For each of the following functions, find an equation of the line tangent to the curve at \(t=t_{0} .\) Choose an orientation for the line that is the same as the direction of \(\mathbf{r}^{\prime}\). $$\mathbf{r}(t)=\langle\sqrt{2 t+1}, \sin \pi t, 4\rangle ; t_{0}=4$$
A pair of nonzero vectors in the plane is linearly dependent if one vector is a scalar multiple of the other. Otherwise, the pair is linearly independent. a. Which pairs of the following vectors are linearly dependent and which are linearly independent: \(\mathbf{u}=\langle 2,-3\rangle\) \(\mathbf{v}=\langle-12,18\rangle,\) and \(\mathbf{w}=\langle 4,6\rangle ?\) b. Geometrically, what does it mean for a pair of nonzero vectors in the plane to be linearly dependent? Linearly independent? c. Prove that if a pair of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is linearly independent, then given any vector \(w\), there are constants \(c_{1}\) and \(c_{2}\) such that \(\mathbf{w}=c_{1} \mathbf{u}+c_{2} \mathbf{v}\)
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