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[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
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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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.
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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