Lab Exercise for Vector Skills
EML 3013C, Dynamic Systems I

Problem 1. Consider the two vectors $\mbox{$\bar{{\bf a}}$}=2\mbox{$\bf{\hat k}$}-4\mbox{$\bf{\hat j}$}+3\mbox{$\bf{\hat i}$}$ and $\mbox{$\bar{{\bf b}}$}=-\mbox{$\bf{\hat j}$}+5\mbox{$\bf{\hat k}$}+\mbox{$\bf{\hat i}$}$ . Compute $\mbox{$\bar{{\bf c}}$}=\mbox{$\bar{{\bf a}}$}\times\mbox{$\bar{{\bf b}}$}$ .

Problem 2. Consider a polar coordinate system with unit vectors as shown below.

$\begin{figure} \begin{center} \includegraphics [height=1.8in, width=2.5in ] {Lab1fig1.eps} \end{center}\end{figure}$

Suppose $\mbox{$\bar{{\bf a}}$}= 5\mbox{${\bf{\hat e}}_z$}-2\mbox{${\bf{\hat e}}_r$}$ and $\mbox{$\bar{{\bf b}}$}= 3\mbox{${\bf{\hat e}}_r$}+ 2\mbox{${\bf{\hat e}}_{\theta}$}$ . Compute $\mbox{$\bar{{\bf c}}$}=\mbox{$\bar{{\bf a}}$}\times\mbox{$\bar{{\bf b}}$}$ .

Problem 3. Suppose $\mbox{$\bar{{\bf u}}$}\times\mbox{$\bar{{\bf v}}$}=0$ . Describe the relationship between the directions of the two vectors.

Problem 4. Suppose $\vert\mbox{$\bar{{\bf u}}$}\times\mbox{$\bar{{\bf v}}$}\vert=\vert\mbox{$\bar{{\bf u}}$}\vert\cdot\vert\mbox{$\bar{{\bf v}}$}\vert$ . Describe the relationship between the directions of the two vectors.

Problem 5. An object rotating about the z axis has the angular velocity vector $\mbox{$\bar{\bf \omega}$}= \omega\mbox{$\bf{\hat k}$}$ and the position of a particle P on the object is given by $\mbox{$\bar{{\bf r}}$}= r_x\mbox{$\bf{\hat i}$}+ r_y\mbox{$\bf{\hat j}$}+ r_z\mbox{$\bf{\hat k}$}$ . The velocity of P is given by $\mbox{$\bar{{\bf v}}$}=\mbox{$\bar{\bf \omega}$}\times\mbox{$\bar{{\bf r}}$}$ . Compute $\mbox{$\bar{{\bf v}}$}$ without using a determinant (i.e., use a distributive rule).

Problem 6. Consider the two vectors $\mbox{$\bar{{\bf a}}$}=2\mbox{$\bf{\hat k}$}-4\mbox{$\bf{\hat j}$}+3\mbox{$\bf{\hat i}$}$ and $\mbox{$\bar{{\bf b}}$}=-\mbox{$\bf{\hat j}$}+5\mbox{$\bf{\hat k}$}+\mbox{$\bf{\hat i}$}$ . Compute $c=\mbox{$\bar{{\bf a}}$}\cdot\mbox{$\bar{{\bf b}}$}$ .

Problem 7. We may write $\mbox{$\bar{{\bf a}}$}\cdot\mbox{$\bar{{\bf b}}$}=\vert\mbox{$\bar{{\bf u}}$}\vert\cdot\vert\mbox{$\bar{{\bf v}}$}\vert$ . Show $\mbox{$\bar{{\bf u}}$}$ and $\mbox{$\bar{{\bf v}}$}$ on the attached diagram. (Hint: The pair (,) is not unique.)

$\begin{figure} \begin{center} \includegraphics [ height=1.8144in, width=2.4517in ] {Lab1fig2.eps} \end{center}\end{figure}$

Problem 8. (a) Suppose $\mbox{$\bar{{\bf u}}$}\cdot\mbox{$\bar{{\bf v}}$}=0$ . What is the relationship between the directions of the two vectors. (b) Suppose $\mbox{$\bar{{\bf u}}$}\cdot\mbox{$\bar{{\bf v}}$}=\vert\mbox{$\bar{{\bf u}}$}\vert\vert\mbox{$\bar{{\bf v}}$}\vert$ What is the relationship between the directions of the two vectors. Make sure you clearly label your answers as (a) or (b).

Problem 9. (a) Express the unit vectors $\mbox{$\bf{\hat n}$}$ and $\mbox{$\bf{\hat \lambda}$}$ in terms of $\mbox{$\bf{\hat i}$}$ and $\mbox{$\bf{\hat j}$}$ as shown in the below figure. (b) What are the x and y components of $\mbox{$\bar{{\bf r}}$}= {\rm 3.0ft}\mbox{$\bf{\hat n}$}- {\rm 1.5ft}\mbox{$\bf{\hat \lambda}$}$ ?

$\begin{figure} \begin{center} \includegraphics [ height=1.8144in, width=2.4517in ] {Lab1fig3.eps} \end{center}\end{figure}$

Problem 10. Write the unit vectors $\mbox{$\bf{\hat i}$}$ and $\mbox{$\bf{\hat j}$}$ corresponding to the (x,y) coordinate system in terms of the unit vectors $\mbox{$\bf{\hat i}$}^\prime$ and $\mbox{$\bf{\hat j}$}^\prime$ corresponding to the $(x^\prime,y^\prime)$ coordinate system. Your answer should be in the form $\mbox{$\bf{\hat i}$}=...$ and $\mbox{$\bf{\hat j}$}=...$ .

$\begin{figure} \begin{center} \includegraphics [ height=1.8144in, width=2.4517in ] {Lab1fig4.eps} \end{center}\end{figure}$

The corresponding postscript file can be downloaded by clicking here. You need Ghostviewer

Back to LABORATORY EXERCISES

For any comments about the contents of this page send e-mail to majura@eng.fsu.edu