Vector Exercise Solutions

EML 3013C: DYNAMIC SYSTEMS I

Lab Exercise for Vector Skills: Solutions

$\begin{displaymath} \bar{a}=3\hat{\imath}-4\hat{\jmath}+2\hat{k}\end{displaymath}$

$\begin{displaymath} \bar{b}=\hat{\imath}-\hat{\jmath}+5\hat{k}\end{displaymath}$

$\begin{displaymath} \bar{c}=\bar{a}\times\bar{b}=\left\vert \begin{array}[c] {c... ...h}-\left( 2-15\right) \hat{\jmath }+\left( -3+4\right) \hat{k}\end{displaymath}$

$\begin{displaymath} \bar{c}=-18\hat{\imath}-13\hat{\jmath}+\hat{k}\end{displaymath}$

$\begin{displaymath} \bar{a}=-2\hat{e}_{r}-0\hat{e}_{\theta}+5\hat{e}_{z}\end{displaymath}$

$\begin{displaymath} \bar{b}=3\hat{e}_{r}+2\hat{e}_{\theta}+0\hat{e}_{z}\end{displaymath}$

$\begin{displaymath} \bar{c}=\bar{a}\times\bar{b}=\left\vert \begin{array}[c] {c... ...( 15-0\right) \hat {e}_{\theta}+\left( -4-0\right) \hat{e}_{z}\end{displaymath}$

$\begin{displaymath} \bar{c}=-10\hat{e}_{r}+15\hat{e}_{\theta}-4\hat{e}_{z}\end{displaymath}$

u and v are parallel

u and v are perpendicular

$\begin{displaymath} \bar{v}=\omega\hat{k}\times\left( r_{x}\hat{\imath}+r_{y}\ha... ...t{k}\right) =\omega r_{x}\hat{\jmath}-\omega r_{y}\hat{\imath} \end{displaymath}$

$\begin{displaymath} \bar{v}=-\omega r_{y}\hat{\imath}+\omega r_{x}\hat{\jmath} \end{displaymath}$

$\begin{displaymath} \bar{a}=3\hat{\imath}-4\hat{\jmath}+2\hat{k}\end{displaymath}$

$\begin{displaymath} \bar{b}=\hat{\imath}-\hat{\jmath}+5\hat{k}\end{displaymath}$

$\begin{displaymath} \bar{c}=\bar{a}.\bar{b}=\ \left( 3\times1\right) +\left( -4\times-1\right) +\left( 2\times5\right) =17 \end{displaymath}$

$\includegraphics [ height=2.156in, width=3.6668in ]{sf1.eps}$

(a) u and v are perpendicular

(b) u and v are parallel

:

(a)	u and v are perpendicular
(b)	u and v are parallel

$\begin{displaymath} \hat{n}=\cos\theta\hat{\imath}+\sin\theta\hat{\jmath} \end{displaymath}$

$\begin{displaymath} \hat{\lambda}=-\sin\theta\hat{\imath}+\cos\theta\hat{\jmath} \end{displaymath}$

$\begin{align*} \bar{r} & =3.0ft\left( \cos\theta\hat{\imath}+\sin\theta\hat{\jma... ...t{\imath}+\left( 3.0\sin\theta-1.5\cos\theta\right) \hat{\jmath}ft \end{align*}$

Problem 10: Coordinate system $\left( x^{\prime},y^{\prime}\right)$ is rotated counterclockwise by angle $\phi$ from the original coordinate system (x,y) therefore the unit vectors in the new coordinate system $(x^{\prime},y^{\prime})$ becomes

$\begin{displaymath} \hat{\imath}^{\prime}=\cos\phi\hat{\imath}+\sin\phi\hat{\jmath}\end{displaymath}$

$\begin{displaymath} \hat{\jmath}^{\prime}=-\sin\phi\hat{\imath}+\cos\phi\hat{\jmath}\end{displaymath}$

$\begin{displaymath} \hat{\imath}=\sin\phi\hat{\imath}^{\prime}-\cos\phi\hat{\jmath}^{\prime}\end{displaymath}$

$\begin{displaymath} \hat{\jmath}=\cos\phi\hat{\imath}^{\prime}+\sin\phi\hat{\jmath}^{\prime}\end{displaymath}$