[cos(255°) -sin (255°) 250] = sin (255°) [20] [249.6* Praya = Terans X Praya cos (255°) –20.6 %3D
[cos(255°) -sin (255°) 250] = sin (255°) [20] [249.6* Praya = Terans X Praya cos (255°) –20.6 %3D
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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I am not sure how this problem is solved, can you show the step-by-step process?
![The equation displayed above describes a coordinate transformation using a transformation matrix and involves a point in a two-dimensional space shifted by translation and rotation. Here is a detailed breakdown of the elements involved:
### Transformation Equation:
**Equation:**
\[ P_{x2,y2} = T_{\text{trans}} \times P_{x3,y3} \]
This equation signifies finding new coordinates \( P_{x2,y2} \) after applying a transformation matrix \( T_{\text{trans}} \) to the original point \( P_{x3,y3} \).
**Transformation Matrix \( T_{\text{trans}} \):**
\[
\begin{bmatrix}
\cos(255^\circ) & -\sin(255^\circ) & 250 \\
\sin(255^\circ) & \cos(255^\circ) & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
- **Rotation:** The matrix includes rotation elements using trigonometric functions (\(\cos\) and \(\sin\)) corresponding to an angle of 255 degrees.
- **Translation:** The matrix includes a translation vector \([250, 0]\) which shifts the point along the x-axis by 250 units.
- **Homogeneous:** The last row \([0, 0, 1]\) is used to maintain the transformation in homogeneous coordinates.
**Original Point \( P_{x3,y3} \):**
\[
\begin{bmatrix}
20 \\
5 \\
1
\end{bmatrix}
\]
- Represents the original coordinates \((20, 5)\) with a homogeneous coordinate \(1\).
### Result:
**New Coordinates:**
\[
\begin{bmatrix}
249.6 \\
-20.6 \\
1
\end{bmatrix}
\]
- After performing the matrix multiplication, the new coordinates are approximately \((249.6, -20.6)\).
### Explanation:
This transformation involves a counterclockwise rotation by 255 degrees and a translation of 250 units in the x-direction. The calculations result in the new position of the point in the coordinate system, reflecting both the rotation and the translation operations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b08af6d-b43b-40f9-8ead-23581b49f3e1%2F65e6d08f-1f14-4121-b8fc-0ee056c2aae7%2Fe5wix3f_processed.png&w=3840&q=75)
Transcribed Image Text:The equation displayed above describes a coordinate transformation using a transformation matrix and involves a point in a two-dimensional space shifted by translation and rotation. Here is a detailed breakdown of the elements involved:
### Transformation Equation:
**Equation:**
\[ P_{x2,y2} = T_{\text{trans}} \times P_{x3,y3} \]
This equation signifies finding new coordinates \( P_{x2,y2} \) after applying a transformation matrix \( T_{\text{trans}} \) to the original point \( P_{x3,y3} \).
**Transformation Matrix \( T_{\text{trans}} \):**
\[
\begin{bmatrix}
\cos(255^\circ) & -\sin(255^\circ) & 250 \\
\sin(255^\circ) & \cos(255^\circ) & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
- **Rotation:** The matrix includes rotation elements using trigonometric functions (\(\cos\) and \(\sin\)) corresponding to an angle of 255 degrees.
- **Translation:** The matrix includes a translation vector \([250, 0]\) which shifts the point along the x-axis by 250 units.
- **Homogeneous:** The last row \([0, 0, 1]\) is used to maintain the transformation in homogeneous coordinates.
**Original Point \( P_{x3,y3} \):**
\[
\begin{bmatrix}
20 \\
5 \\
1
\end{bmatrix}
\]
- Represents the original coordinates \((20, 5)\) with a homogeneous coordinate \(1\).
### Result:
**New Coordinates:**
\[
\begin{bmatrix}
249.6 \\
-20.6 \\
1
\end{bmatrix}
\]
- After performing the matrix multiplication, the new coordinates are approximately \((249.6, -20.6)\).
### Explanation:
This transformation involves a counterclockwise rotation by 255 degrees and a translation of 250 units in the x-direction. The calculations result in the new position of the point in the coordinate system, reflecting both the rotation and the translation operations.
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