Chapter 11

Let \(\quad \mathbf{a}=-3 \mathbf{i}+2 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{b}=-\mathbf{i}+2 \mathbf{j}-4 \mathbf{k}, \quad\) and \(\mathbf{c}=7 \mathbf{i}+3 \mathbf{j}-4 \mathbf{k} .\) Find each of the following: (a) \(\mathbf{a} \times \mathbf{b}\) (b) \(\mathbf{a} \times(\mathbf{b}+\mathbf{c})\) (c) \(\mathbf{a} \cdot(\mathbf{b}+\mathbf{c})\) (d) \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})\)

Name and sketch the graph of each of the following equations in three-space. $$ 4 x^{2}+36 y^{2}=144 $$

Let \(\mathbf{a}=-2 \mathbf{i}+3 \mathbf{j}, \mathbf{b}=2 \mathbf{i}-3 \mathbf{j}\), and \(\mathbf{c}=-5 \mathbf{j}\). Find each of the following: (a) \(2 \mathbf{a}-4 \mathbf{b}\) (b) \(\mathbf{a} \cdot \mathbf{b}\) (c) \(\mathbf{a} \cdot(\mathbf{b}+\mathbf{c})\) (d) \((-2 \mathbf{a}+3 \mathbf{b}) \cdot 5 \mathbf{c}\) (e) \(\|\mathbf{a}\| \mathbf{c} \cdot \mathbf{a}\) (f) \(\mathbf{b} \cdot \mathbf{b}-\|\mathbf{b}\|\)

Find the parametric equations of the line through the given pair of points. \((1,-2,3),(4,5,6)\)

In Problems \(1-8\), find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow 1}\left[2 t \mathbf{i}-t^{2} \mathbf{j}\right] $$

In Problems 1-6, sketch the curve over the indicated domain for \(t\). Find \(\mathbf{v}, \mathbf{a}, \mathbf{T}\), and \(\kappa\) at the point where \(t=t_{1} .\) $$ \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j} ; \quad 0 \leq t \leq 2 ; t_{1}=1 $$

sketch the curve over the indicated domain for \(t\). Find \(\mathbf{v}, \mathbf{a}, \mathbf{T}\), and \(\kappa\) at the point where \(t=t_{1} .\) $$ \mathbf{r}(t)=t^{2} \mathbf{i}+(2 t+1) \mathbf{j} ; \quad 0 \leq t \leq 2 ; t_{1}=1 $$

Find the parametric equations of the line through the given pair of points. \((2,-1,-5),(7,-2,3)\)

Name and sketch the graph of each of the following equations in three-space. $$ y^{2}+z^{2}=15 $$

Change the following from cylindrical to spherical coordinates. (a) \((1, \pi / 2,1)\) (b) \((-2, \pi / 4,2)\)