Chapter 10: Problem 23
In your own words, explain the difference between the velocity of an object and its speed.
Chapter 10: Problem 23
In your own words, explain the difference between the velocity of an object and its speed.
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Evaluate the definite integral. $$ \int_{-1}^{1}\left(t \mathbf{i}+t^{3} \mathbf{j}+\sqrt[3]{t} \mathbf{k}\right) d t $$
Consider a particle moving on a circular path of radius \(b\) described by $$ \begin{aligned} &\mathbf{r}(t)=b \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\\\ &\text { where } \omega=d \theta / d t \text { is the constant angular velocity. } \end{aligned} $$ $$ \text { Find the velocity vector and show that it is orthogonal to } \mathbf{r}(t) $$
In Exercises \(\mathbf{3 7}\) and \(\mathbf{3 8 ,}\) find (a) \(\quad D_{t}[\mathbf{r}(t) \cdot \mathbf{u}(t)] \quad\) and (b) \(D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]\) by differentiating the product, then applying the properties of Theorem 10.2. $$ \mathbf{r}(t)=t \mathbf{i}+2 t^{2} \mathbf{j}+t^{3} \mathbf{k}, \quad \mathbf{u}(t)=t^{4} \mathbf{k} $$
The graph of the vector-valued function \(\mathbf{r}(t)\) and a tangent vector to the graph at \(t=t_{0}\) are given. (a) Find a set of parametric equations for the tangent line to the graph at \(t=t_{0}\) (b) Use the equations for the tangent line to approximate \(\mathbf{r}\left(t_{0}+\mathbf{0 . 1}\right)\) $$ \mathbf{r}(t)=\left\langle t,-t^{2}, \frac{1}{4} t^{3}\right\rangle, \quad t_{0}=1 $$
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