# (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?
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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) $$\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}$$
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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.
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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.
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Problem: find acondition on H, involving $$\displaystyle{P}_{{0}}$$ and k, that will prevent solutions from growing exponentially.
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