91Ó°ÊÓ

An object is acted upon by the forces \(\mathbf{F}_{1}=\langle 10,6,3\rangle\) and \(\mathbf{F}_{2}=\langle 0,4,9\rangle .\) Find the force \(\mathbf{F}_{3}\) that must act on the object so that the sum of the forces is zero.

Find the domain of the following vector-valued functions. $$\mathbf{r}(t)=\frac{2}{t-1} \mathbf{i}+\frac{3}{t+2} \mathbf{j}$$

Find the time of flight, range, and maximum height of the following two- dimensional trajectories, assuming no forces other than gravity. In each case, the initial position is \langle 0,0\rangle and the initial velocity is \(\mathbf{v}_{0}=\left\langle u_{0}, v_{0}\right\rangle\). $$\left\langle u_{0}, v_{0}\right\rangle=\langle 40,80\rangle \mathrm{m} / \mathrm{s}$$

Determine whether the following statements are true and give an explanation or counterexample. a. Suppose \(\mathbf{u}\) and \(\mathbf{v}\) both make a \(45^{\circ}\) angle with \(\mathbf{w}\) in \(\mathrm{R}^{3}\). Then \(\mathbf{u}+\mathbf{v}\) makes a \(45^{\circ}\) angle with \(\mathbf{w}\) b. Suppose \(\mathbf{u}\) and \(\mathbf{v}\) both make a \(90^{\circ}\) angle with \(\mathbf{w}\) in \(\mathbb{R}^{3}\). Then \(\mathbf{u}+\mathbf{v}\) can never make a \(90^{\circ}\) angle with \(\mathbf{w}\) c. \(i+j+k=0\) d. The intersection of the planes \(x=1, y=1,\) and \(z=1\) is a point.

Find the time of flight, range, and maximum height of the following two- dimensional trajectories, assuming no forces other than gravity. In each case, the initial position is \langle 0,0\rangle and the initial velocity is \(\mathbf{v}_{0}=\left\langle u_{0}, v_{0}\right\rangle\). $$\text { Initial speed }\left|\mathbf{v}_{0}\right|=400 \mathrm{ft} / \mathrm{s}, \text { launch angle } \alpha=60^{\circ}$$

Find the length of the entire spiral \(r=e^{-a \theta},\) for \(\theta \geq 0\) and \(a>0\)

Find the area of the following triangles \(T\) The vertices of \(T\) are \(O(0,0,0), P(1,2,3),\) and \(Q(6,5,4)\)

Find the domain of the following vector-valued functions. $$\mathbf{r}(t)=\sqrt{t+2} \mathbf{i}+\sqrt{2-t} \mathbf{j}$$

Vectors with equal projections Given a fixed vector \(\mathbf{v},\) there is an infinite set of vectors \(\mathbf{u}\) with the same value of \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). Let \(\mathbf{v}=\langle 0,0,1\rangle .\) Give a description of all position vectors \(\mathbf{u}\) such that \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\) \(=\operatorname{proj}_{\mathbf{v}}\langle 1,2,3\rangle\).

Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\left\langle e^{2 t}, 1-2 e^{-t}, 1-2 e^{t}\right\rangle ; \mathbf{r}(0)=\langle 1,1,1\rangle$$