91Ó°ÊÓ

Explain how to decompose the acceleration vector of a moving object into its tangential and normal components.

Explain what it means for a curve to be parameterized by its arc length.

Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the following parameterized curves. $$\mathbf{r}(t)=\left\langle\int_{0}^{t} \cos \left(\pi u^{2} / 2\right) d u, \int_{0}^{t} \sin \left(\pi u^{2} / 2\right) d u\right\rangle, t>0$$

Refer to the figure and carry out the following vector operations. Vector addition Write the following vectors as sums of scalar multiples of \(\mathbf{u}\) and \(\mathbf{v}\) a. \(\overrightarrow{O E}\) b. \(\overrightarrow{O B}\) c. \(\overrightarrow{O F}\) d. \(\overrightarrow{O G}\) e. \(\overrightarrow{O C}\) f. \(\overrightarrow{O I}\) g. \(\overrightarrow{O J}\) h. \(\overrightarrow{O K}\) i. \(\overrightarrow{O L}\)

Sketch the plane parallel to the \(y z\) -plane through (2,4,2) and find its equation.

Use the alternative curvature formula \(\kappa=|\mathbf{v} \times \mathbf{a}| /|\mathbf{v}|^{3}\) to find the curvature of the following parameterized curves. $$\mathbf{r}(t)=\langle\sqrt{3} \sin t, \sin t, 2 \cos t\rangle$$

Find an equation of the line segment joining the first point to the second point. $$(1,0,1) \text { and }(0,-2,1)$$

Determine whether the following curves use arc length as a parameter. If not, find a description that uses arc length as a parameter. $$\mathbf{r}(t)=\left\langle\frac{t}{\sqrt{3}}, \frac{t}{\sqrt{3}}, \frac{t}{\sqrt{3}}\right\rangle, \text { for } 0 \leq t \leq 10$$

Parallel and normal forces Find the components of the vertical force \(\mathbf{F}=\langle 0,-10\rangle\) in the directions parallel to and normal to the following inclined planes. Show that the total force is the sum of the two component forces. A plane that makes an angle of \(\pi / 6\) with the positive \(x\) -axis

Assume that \(f\) is twice differentiable. Prove that the curve \(y=f(x)\) has curvature $$\kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left(1+f^{\prime}(x)^{2}\right)^{3 / 2}}$$. (Hint: Use the parametric description \(x=t, y=f(t) .\) )