Determining Image Coordinates

Oct 13, 2025 by windailys.com 8 views

Introduction

In mathematics, particularly in geometry and linear algebra, rotations are an essential concept used to describe the transformation of objects in a two-dimensional or three-dimensional space. A rotation is a transformation that turns a figure around a fixed point called the center of rotation. In this article, we will explore how to determine the coordinates of the image of a point T(-2, 9) under a specific rotation.

Understanding Rotations

A rotation is a transformation that turns a figure around a fixed point called the center of rotation. The center of rotation remains fixed, and the figure is rotated around it. The amount of rotation is measured in degrees, and it can be clockwise or counterclockwise.

Rotation Matrix

A rotation matrix is a square matrix used to describe a rotation in a two-dimensional or three-dimensional space. The rotation matrix is given by:

R(\theta) = \begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}

where θ is the angle of rotation.

Determining Image Coordinates

To determine the image coordinates of a point T(-2, 9) under a rotation, we need to multiply the rotation matrix by the coordinates of the point.

Example 1: Rotation by 90° Counterclockwise

In this example, we will rotate the point T(-2, 9) by 90° counterclockwise.

R(90°) = \begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}

The coordinates of the point T(-2, 9) are given by:

T = \begin{bmatrix}
-2 \\
9
\end{bmatrix}

To determine the image coordinates of the point T under the rotation, we multiply the rotation matrix by the coordinates of the point:

R(90°) \cdot T = \begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix} \cdot \begin{bmatrix}
-2 \\
9
\end{bmatrix} = \begin{bmatrix}
-9 \\
-2
\end{bmatrix}

Therefore, the image coordinates of the point T(-2, 9) under the rotation by 90° counterclockwise are (-9, -2).

Example 2: Rotation by 180° Counterclockwise

In this example, we will rotate the point T(-2, 9) by 180° counterclockwise.

R(180°) = \begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix}

The coordinates of the point T(-2, 9) are given by:

T = \begin{bmatrix}
-2 \\
9
\end{bmatrix}

To determine the image coordinates of the point T under the rotation, we multiply the rotation matrix by the coordinates of the point:

R(180°) \cdot T = \begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix} \cdot \begin{bmatrix}
-2 \\
9
\end{bmatrix} = \begin{bmatrix}
2 \\
-9
\end{bmatrix}

Therefore, the image coordinates of the point T(-2, 9) under the rotation by 180° counterclockwise are (2, -9).

Example 3: Rotation by 270° Counterclockwise

In this example, we will rotate the point T(-2, 9) by 270° counterclockwise.

R(270°) = \begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}

The coordinates of the point T(-2, 9) are given by:

T = \begin{bmatrix}
-2 \\
9
\end{bmatrix}

To determine the image coordinates of the point T under the rotation, we multiply the rotation matrix by the coordinates of the point:

R(270°) \cdot T = \begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix} \cdot \begin{bmatrix}
-2 \\
9
\end{bmatrix} = \begin{bmatrix}
9 \\
2
\end{bmatrix}

Therefore, the image coordinates of the point T(-2, 9) under the rotation by 270° counterclockwise are (9, 2).

Conclusion

Introduction

In our previous article, we explored how to determine the coordinates of the image of a point T(-2, 9) under specific rotations. We used rotation matrices to describe the rotations and multiplied the rotation matrices by the coordinates of the point to determine the image coordinates. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on this topic.

Q: What is a rotation matrix?

A: A rotation matrix is a square matrix used to describe a rotation in a two-dimensional or three-dimensional space. The rotation matrix is given by:

R(\theta) = \begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}

where θ is the angle of rotation.

Q: How do I determine the image coordinates of a point under a rotation?

A: To determine the image coordinates of a point under a rotation, you need to multiply the rotation matrix by the coordinates of the point.

Q: What is the difference between a clockwise and counterclockwise rotation?

A: A clockwise rotation is a rotation that turns a figure in a clockwise direction, while a counterclockwise rotation is a rotation that turns a figure in a counterclockwise direction.

Q: How do I determine the angle of rotation?

A: The angle of rotation is the amount of rotation measured in degrees. You can determine the angle of rotation by using the rotation matrix and the coordinates of the point.

Q: Can I use a rotation matrix to describe a rotation in three-dimensional space?

A: Yes, you can use a rotation matrix to describe a rotation in three-dimensional space. However, the rotation matrix will be a 3x3 matrix instead of a 2x2 matrix.

Q: How do I apply a rotation to a point in three-dimensional space?

A: To apply a rotation to a point in three-dimensional space, you need to multiply the rotation matrix by the coordinates of the point.

Q: What is the center of rotation?

A: The center of rotation is the point around which the rotation takes place. The center of rotation remains fixed during the rotation.

Q: Can I use a rotation matrix to describe a rotation that is not a multiple of 90°?

A: Yes, you can use a rotation matrix to describe a rotation that is not a multiple of 90°. However, the rotation matrix will be a more complex matrix.

Q: How do I determine the image coordinates of a point under a rotation that is not a multiple of 90°?

A: To determine the image coordinates of a point under a rotation that is not a multiple of 90°, you need to use a more complex rotation matrix and multiply it by the coordinates of the point.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts and provide additional information on determining the coordinates of the image of a point under specific rotations. We hope this article has been helpful in understanding this concept.

Additional Resources

Frequently Asked Questions

Q: What is a rotation matrix? A: A rotation matrix is a square matrix used to describe a rotation in a two-dimensional or three-dimensional space.
Q: How do I determine the image coordinates of a point under a rotation? A: To determine the image coordinates of a point under a rotation, you need to multiply the rotation matrix by the coordinates of the point.
Q: What is the difference between a clockwise and counterclockwise rotation? A: A clockwise rotation is a rotation that turns a figure in a clockwise direction, while a counterclockwise rotation is a rotation that turns a figure in a counterclockwise direction.