The question is to give explicit Affine transformation taking any one of these 3 parallelograms to another. **Text Transcription:**Let \( A = \begin{pmatrix} 0 \\ 0 \end{pmatrix}, B = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \) and \( C = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \) be points in \(\Pi\). Define the points\[ D_1 = A + B - C = \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \quad D_2 = C + A - B = \begin{pmatrix} -1 \\ 1 \end{pmatrix}, \quad D_3 = B + C - A = \begin{pmatrix} 1 \\ 1 \end{pmatrix}. \]Give explicit invertible linear maps relating any two of the three parallelograms \( ABCD_1, ABCD_2, \) and \( ABCD_3 \). ### Coordinate System and GeometryThis image depicts a geometric diagram within a coordinate plane. It features a set of labeled points and vectors, illustrating the relationships between these points.#### Points and Coordinates:- **A** is located at the origin with coordinates \( A = (0, 0) \).- **B** is on the x-axis with coordinates \( B = (1, 0) \).- **C** is on the y-axis with coordinates \( C = (0, 1) \).- **D_1** is in the fourth quadrant with coordinates \( D_1 = (-1, -1) \).- **D_2** is in the second quadrant with coordinates \( D_2 = (-1, 1) \).- **D_3** is in the first quadrant with coordinates \( D_3 = (1, 1) \).#### Diagram Description:- The figure includes a triangle highlighted in red.- The triangle is formed by the points A, B, and D_3.- Lines or vectors are drawn from the origin to points B, C, D_1, and D_2.- The lines connecting these points form a structure that resembles two adjacent triangles sharing a side.This diagram can be used to understand vector addition, the properties of triangles in a plane, or transformations involving reflections or rotations in geometry.

Answered: Image /qna-images/answer/23b8472e-0fe1-42e7-a0ea-580047819ec4.jpg

Let A = (6), B = (,), and C C) be points in II. Define the points %3D D, 3D A + B — С %3 = (_;), D2 = C + A – B = (), D3 = B + C – A = | | Give explicit invertible linear maps relating any two of the three parallelograms АВCD,АBCD2 and ABCD3-

Let A = (6), B = (,), and C C) be points in II. Define the points %3D D, 3D A + B — С %3 = (_;), D2 = C + A – B = (), D3 = B + C – A = | | Give explicit invertible linear maps relating any two of the three parallelograms АВCD,АBCD2 and ABCD3-

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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Free Body Diagram

The system of forces acting on a body tends to either rotate it or make the body undergo motion. In general, when an external agent exerts a force on a body, the body undergoes translational motion and rotational motion. The body has six degrees of freedom under the influence of that force. Here, three degrees of freedom are from the translational motion while the other three are from the rotation motion. However, sometimes the bodies are not allowed to undergo any motion and are fixed, under such conditions the body is said to be constrained.

Question

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The question is to give explicit Affine transformation taking any one of these
3 parallelograms to another.

**Text Transcription:**

Let \( A = \begin{pmatrix} 0 \\ 0 \end{pmatrix}, B = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \) and \( C = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \) be points in \(\Pi\). Define the points

\[ D_1 = A + B - C = \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \quad D_2 = C + A - B = \begin{pmatrix} -1 \\ 1 \end{pmatrix}, \quad D_3 = B + C - A = \begin{pmatrix} 1 \\ 1 \end{pmatrix}. \]

Give explicit invertible linear maps relating any two of the three parallelograms \( ABCD_1, ABCD_2, \) and \( ABCD_3 \).

### Coordinate System and Geometry

This image depicts a geometric diagram within a coordinate plane. It features a set of labeled points and vectors, illustrating the relationships between these points.

#### Points and Coordinates:
- **A** is located at the origin with coordinates \( A = (0, 0) \).
- **B** is on the x-axis with coordinates \( B = (1, 0) \).
- **C** is on the y-axis with coordinates \( C = (0, 1) \).
- **D_1** is in the fourth quadrant with coordinates \( D_1 = (-1, -1) \).
- **D_2** is in the second quadrant with coordinates \( D_2 = (-1, 1) \).
- **D_3** is in the first quadrant with coordinates \( D_3 = (1, 1) \).

#### Diagram Description:
- The figure includes a triangle highlighted in red.
- The triangle is formed by the points A, B, and D_3.
- Lines or vectors are drawn from the origin to points B, C, D_1, and D_2.
- The lines connecting these points form a structure that resembles two adjacent triangles sharing a side.

This diagram can be used to understand vector addition, the properties of triangles in a plane, or transformations involving reflections or rotations in geometry.

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