# (a) The new coordinates geometrically if X represents the point (0, sqrt{2}) m and this point is rotated about the origin 45^{circ} clockwise and then translated 2 units to the right and 3 units upward. (b) The value of Y = ABX, and explain the result. (c) If ABX equal to BAX. Interpret the resul. (d) A matrix that translate Y back to X

Question
Transformation properties
(a) The new coordinates geometrically if X represents the point $$(0,\ \sqrt{2})$$ m and this
point is rotated about the origin $$45^{\circ}$$ clockwise and then translated 2 units to the right and 3 units upward.
(b) The value of $$Y = ABX,$$ and explain the result.
(c) If ABX equal to BAX. Interpret the resul.
(d) A matrix that translate Y back to X

2021-02-20
Given:
The matrices
$$A=\begin{bmatrix}1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{bmatrix},\ B=\begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
Concept used:
When a point is rotated an angle theta clockwise about the origin, the transforming matrix is
$$\begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
To translate a point (x, y) horizontallyh units and vertically k units, we use the transformation matrix
$$\begin{bmatrix} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{bmatrix}$$
Calculation:
(a) Let the point $$(0,\ \sqrt{2})$$ be represented by column matrix
$$X=\begin{bmatrix} 1 \\ \sqrt{2} \\ 1 \end{bmatrix}$$
When it is rotated $$45^{\circ}$$ about the origin, then we have
$$\begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 0 \\ \sqrt{2} \\ 1 \end{bmatrix}=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$$
So after rotating $$45^{\circ}$$ about the origin, the new coordinates are (1, 1).
Now we need to translate this point 2 units to the right and 3 units upward, so using above concept we have
$$\begin{bmatrix}1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}=\begin{bmatrix} 3 \\ 4 \\ 1 \end{bmatrix}$$
Hence after rotating $$45^{\circ}$$ about origin and then translating 2 units to the right and 3 units upward, we reach to the point (3, 4).
(b) Now to compure $$Y = ABX,$$ we have
$$Y=ABX=\begin{bmatrix}1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ 1 & 0 & 1 \end{bmatrix}\begin{bmatrix} 0 \\ \sqrt{2} \\ 1 \end{bmatrix}$$
$$=\begin{bmatrix}1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$$
$$=\begin{bmatrix} 3 \\ 4 \\ 1 \end{bmatrix}$$
It is same as the above transformations.
Hence the first matrix A represents the translation of 2 units the right and 3 units upward.
And the matrix B represent the rotation of $$45^{\circ}$$ clockwise about the origin of the point X.
(c) Now to check if ABX equal to BAX, we compute BAX first. So we have
$$BAX=\begin{bmatrix}\frac{1}{\sqrt{1}} & \frac{1}{\sqrt{2}} & 0 \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix}2 \\ 3\ +\ \sqrt{2} \\ 1 \end{bmatrix}\begin{bmatrix}\frac{5\ +\ \sqrt{2}}{\sqrt{2}} \\ \frac{1\ +\ \sqrt{2}}{\sqrt{2}} \\ 1 \end{bmatrix}$$
We can see that ABX is not equal to BAX. It implies that if we translate the point X, 2 units to the right and 3 units upward first and then rotate about the origin $$45^{\circ}$$ clockwise we will not reach to the same point as we get by first rotating about the origin and then translating.
(d) To find the matrix that translate Y back to X, we use the concept
$$Y = ABX$$
$$\Rightarrow\ (AB)^{-1} Y = X$$
Now we first find $$(AB)^{-1} = B^{-1} A^{-1},$$ so we get
$$(AB)^{-1} = B^{-1} A^{-1},$$
$$\begin{bmatrix}\frac{1}{\sqrt{1}} & -\frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix}1 & 0 & -2 \\ 0 & 1 & -3 \\ 0 & 0 & 1 \end{bmatrix}$$
Thus, the matrix that translate Y back
$$=\begin{bmatrix}\frac{1}{\sqrt{1}} & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & -\frac{5}{\sqrt{2}} \\ 0 & 0 & 1 \end{bmatrix}$$
to X, is given by $$\begin{bmatrix}\frac{1}{\sqrt{1}} & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & -\frac{5}{\sqrt{2}} \\ 0 & 0 & 1 \end{bmatrix}$$
Conclusion:
Using the properties of matrix multiplication, we found the rotation and translation of the point X.

### Relevant Questions

a) To find:
The images of the following points under under a 90^circ rotation counterclockwise about the origin:
I. $$(2,\ 3)$$
II. $$(-1,\ 2)$$
III, (m,n) interms of m and n
b)To show:
That under a half-turn with the origin as center, the image of a point $$(a,\ b)\ \text{has coordinates}\ (-a,\ -b).$$
c) To find:
The image of $$P (a,\ b)\ text{under the rotation clockwise by} 90^{\circ}$$ about the origin.
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
The crane shown in the drawing is lifting a 182-kg crate upward with an acceleration of $$\displaystyle{1.5}\frac{{m}}{{s}^{{2}}}$$. The cable from the crate passes over a solid cylindrical pulley at the top of the boom. The pulley has a mass of 130 kg. The cable is then wound ontoa hollow cylindrical drum that is mounted on the deck of the crane.The mass of the drum is 150 kg, and its radius is 0.76 m. The engine applies a counter clockwise torque to the drum in order towind up the cable. What is the magnitude of this torque? Ignore the mass of the cable.
The graph of y = f(x) contains the point (0,2), $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{-{x}}}{{{y}{e}^{{{x}^{{2}}}}}}}$$, and f(x) is greater than 0 for all x, then f(x)=
A) $$\displaystyle{3}+{e}^{{-{x}^{{2}}}}$$
B) $$\displaystyle\sqrt{{{3}}}+{e}^{{-{x}}}$$
C) $$\displaystyle{1}+{e}^{{-{x}}}$$
D) $$\displaystyle\sqrt{{{3}+{e}^{{-{x}^{{2}}}}}}$$
E) $$\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}$$
We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:
a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.
A helicopter carrying dr. evil takes off with a constant upward acceleration of $$\displaystyle{5.0}\ \frac{{m}}{{s}^{{2}}}$$. Secret agent austin powers jumps on just as the helicopter lifts off the ground. Afterthe two men struggle for 10.0 s, powers shuts off the engineand steps out of the helicopter. Assume that the helicopter is infree fall after its engine is shut off and ignore effects of airresistance.
a) What is the max height above ground reached by the helicopter?
b) Powers deploys a jet pack strapped on his back 7.0s after leaving the helicopter, and then he has a constant downward acceleration with magnitude 2.0 m/s2. how far is powers above the ground when the helicopter crashes into the ground.
An electron is fired at a speed of $$\displaystyle{v}_{{0}}={5.6}\times{10}^{{6}}$$ m/s and at an angle of $$\displaystyle\theta_{{0}}=–{45}^{\circ}$$ between two parallel conductingplates that are D=2.0 mm apart, as in Figure. Ifthe potential difference between the plates is $$\displaystyle\triangle{V}={100}\ {V}$$, determine (a) how close d the electron will get to the bottom plate and (b) where the electron will strike the top plate.
1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance.
Problem: find acondition on H, involving $$\displaystyle{P}_{{0}}$$ and k, that will prevent solutions from growing exponentially.