Chapter 11: Problem 4
What position vector is equal to the vector from (3,5,-2) to (0,-6,3)\(?\)
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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 \(\mathbf{r}^{\prime}\). $$\mathbf{r}(t)=\left\langle 3 t-1,7 t+2, t^{2}\right\rangle ; t_{0}=1$$
Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle e^{4 t}, 2 e^{-4 t}+1,2 e^{-t}\right\rangle$$
Diagonals of a parallelogram Consider the parallelogram with adjacent sides \(\mathbf{u}\) and \(\mathbf{v}\) a. Show that the diagonals of the parallelogram are \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\) b. Prove that the diagonals have the same length if and only if \(\mathbf{u} \cdot \mathbf{v}=0\) c. Show that the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides.
Proof of Sum Rule By expressing \(\mathbf{u}\) and \(\mathbf{v}\) in terms of their components, prove that $$\frac{d}{d t}(\mathbf{u}(t)+\mathbf{v}(t))=\mathbf{u}^{\prime}(t)+\mathbf{v}^{\prime}(t)$$
Prove that \(|c \mathbf{v}|=|c||\mathbf{v}|,\) where \(c\) is a scalar and \(\mathbf{v}\) is a vector.
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