For solutions of $\overrightarrow{\textbf{x}}'=\textbf{A}\overrightarrow{\textbf{x}}$ with A skew-symmetric, the length of vector $\overrightarrow{\textbf{x}}$ is constant. What does this mean in terms of phase-plane trajectories? Use IDE's Matrix Element Input tool, or an open-ended graphic DE solver, to verify this for the following systems, and explain the role of the parameter k. (a)

$$\overrightarrow{\textbf{x}}'=\begin{bmatrix}0&1\\-1&0\end{bmatrix}\overrightarrow{\textbf{x}}$$

(b)

$$\overrightarrow{\textbf{x}}'=\begin{bmatrix}0&k\\-k&0\end{bmatrix}\overrightarrow{\textbf{x}},k>0$$

.

8. in the following diagram of a plant cell, the small circles represent water molecules. Which statement best describes the behavior of

‘most of these water molecules?

(1) They move from region A to region B.

(2) They move from region B to region A.

(3) They do not move in either direction.

(4) Their overall movement is equal in both direstiont:

Identify each of the following as the formula mass, a mole, or the molar mass of a carbon monoxide (CO) molecule. One item does not make sense. Identify it. a. 28.01 grams b. 12.01 grams c. 28.01 amu d. 6.02 x 1023 molecules

The Ungers have an adjusted gross income of $117,445. They are looking at a new house that would carry a monthly mortgage payment of$1,877. Their annual property taxes would be $6,780, and their semi-annual homeowner’s premium would be$710. Find their front-end ratio to the nearest percent.

Write the value of the underlined digit.

$$9,\underline{4}22,850$$

The infrastructure needs of a region (for example, the number of miles of electrical cable, the number of miles of roads, the number of gas stations) depend on its population.Cities enjoy economies of scale.For example, the number of gas stations is proportional to the population raised to the power of 0. 77.If city A is 10 times bigger than city B, how do their number of gas stations compare?

Solve $u_{x x}+u_{y y}=0$ in the wedge $r<a, 0<\theta<\beta$ with the $\mathrm{BCs}$ $u=\theta \quad$ on $r=a, \quad u=0 \quad$ on $\theta=0, \quad$ and $\quad u=\beta \quad$ on $\theta=\beta$

Verify the relationships

$$f \left( f ^ { - 1 } ( x ) \right) = x$$

and

$$f ^ { - 1 } ( f ( x ) ) = x$$

. f(x)=

$$= \frac { 1 } { 4 } x$$

+1

Examine the effects of applying one translation followed by another. This two-step transformation is called a composition of translations. a. Draw $\triangle A B C$ with vertex coordinates A(1, 5), B(6, 2), C(8, 11). b. Translate $\triangle A B C$ using the translation with horizontal component 6 and vertical component -10. Then apply the translation with horizontal component -8 and vertical component 2 to the image of $\triangle A B C$. What were the net horizontal and vertical distances that you translated the figure and in which directions? c. Compare the size, shape, and orientation of $\triangle A B C$ and the final image. d. Compare the coordinates of the vertices of the final image with those of $\triangle A B C$. What pattern do you observe? e. Could you apply a single transformation to $\triangle A B C$ and obtain the same final image? If so, describe that transformation as completely as possible. f. Repeat Parts a–e for a new figure and two new translations of your choice. Keep a record of the coordinates of the vertices of your figure and its final image. Also, record the components of your translations. g. Look back at your work in Parts a–f. Suppose the following two translations are applied in succession.

$$\begin{array} { l } { ( x , y ) \rightarrow ( x + a , y + b ) } \\ { ( x , y ) \rightarrow ( x + h , y + k ) } \end{array}$$

Write a symbolic rule (x, y) $\rightarrow$ (__, __) which describes the new combined transformation. Compare your rule with that of others and resolve any differences.

a. Find the de Broglie wavelength of a proton (mass $1.67 \times 10 ^ { - 27 } \mathrm { kg })$ whose kinetic energy is 200.0 MeV. b. What is the de Broglie wavelength of an electron in the n = 2 state of hydrogen?

If people always spoke as directly as possible and never used euphemisms or abstractions to gloss over facts, what do you think would be the results? Be specific in your answer.

What might the witches' predictions about Banquo mean?

Alison often sketches her quilt designs on a coordinate grid. The coordinates for a section of one her designs are A(-4,6) , B(-2,-2) , C(0,0) , D(1,-1) , and E(3,6). She wants to transform the original design by a horizontal stretch about the y-axis by a factor of 2, a reflection in the x-axis, and a translation of 4 units up and 3 units to the left.

a) Determine the coordinates of the image points, A' , B' , C' , D' and E'.

b) If the original design was defined by the function y=f(x), determine the equation of the design resulting from the transformations.

Which effects of the Bantus-peaking migrations do you think had the most long-term impact? Explain.