91Ó°ÊÓ

Let \(\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,\) and \(\mathbf{w}=\langle-1,0\rangle .\) Carry out the following computations. Find \(|-2 \mathbf{v}|\)

Give a geometric description of the following sets of points. $$x^{2}+y^{2}+z^{2}-8 x-14 y-18 z \leq 79$$

Graph the curves described by the following functions, indicating the direction of positive orientation. Try to anticipate the shape of the curve before using a graphing utility. $$\mathbf{r}(t)=e^{-t / 20} \sin t \mathbf{i}+e^{-t / 20} \cos t \mathbf{j}+t \mathbf{k}, \text { for } 0 \leq t < \infty$$

For the given vectors \(\mathbf{u}\) and \(\mathbf{v},\) calculate proj\(_{\mathbf{v}} \mathbf{u}\) and \(\operatorname{scal}_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u}=5 \mathbf{i}+\mathbf{j}-5 \mathbf{k}\) and \(\mathbf{v}=-\mathbf{i}+\mathbf{j}-2 \mathbf{k}\)

Arc length of polar curves Find the length of the following polar curves. The complete cardioid \(r=4+4 \sin \theta\)

Find a vector normal to the given vectors. $$\langle 0,1,2\rangle \text { and }\langle-2,0,3\rangle$$

Given an acceleration vector, initial velocity $\left\langle u_{0}, v_{0}\right\rangle,\( and initial position \)\left\langle x_{0}, y_{0}\right\rangle,\( find the velocity and position vectors, for \)t \geq 0$. $$\mathbf{a}(t)=\langle\cos t, 2 \sin t\rangle,\left\langle u_{0}, v_{0}\right\rangle=\langle 0,1\rangle,\left\langle x_{0}, y_{0}\right\rangle=\langle 1,0\rangle$$

Given an acceleration vector, initial velocity $\left\langle u_{0}, v_{0}\right\rangle,\( and initial position \)\left\langle x_{0}, y_{0}\right\rangle,\( find the velocity and position vectors, for \)t \geq 0$. $$\mathbf{a}(t)=\left\langle e^{-t}, 1\right\rangle,\left\langle u_{0}, v_{0}\right\rangle=\langle 1,0\rangle,\left\langle x_{0}, y_{0}\right\rangle=\langle 0,0\rangle$$

Arc length of polar curves Find the length of the following polar curves. The spiral \(r=4 \theta^{2},\) for \(0 \leq \theta \leq 6\)

Let \(\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,\) and \(\mathbf{w}=\langle-1,0\rangle .\) Carry out the following computations. Find \(|\mathbf{u}+\mathbf{v}+\mathbf{w}|\)