Point A Has Coordinates { A (3,2) $}$. The Point Is Rotated { 180^{\circ} $}$ Clockwise Around The Origin.Find The Coordinates Of Point { A^{\prime} $}$.A. { A^{\prime}(2,3) $}$ B. [$ A^{\prime}(-3,-2)

Mar 9, 2025 by ADMIN 203 views

**Rotating Points in the Coordinate Plane: A Step-by-Step Guide**

Introduction

In mathematics, rotating points in the coordinate plane is a fundamental concept that has numerous applications in various fields, including geometry, trigonometry, and physics. In this article, we will explore the process of rotating a point around the origin in the coordinate plane and provide a step-by-step guide on how to find the coordinates of the rotated point.

What is Rotation in the Coordinate Plane?

Rotation in the coordinate plane refers to the process of rotating a point or a shape around a fixed point, known as the origin. The origin is the point where the x-axis and y-axis intersect. When a point is rotated around the origin, its coordinates change, and the new coordinates can be found using specific formulas.

Rotating a Point Clockwise Around the Origin

To rotate a point clockwise around the origin, we use the following formula:

x' = -x y' = -y

where (x, y) are the original coordinates of the point, and (x', y') are the new coordinates after rotation.

Example: Rotating Point A

Let's consider an example to illustrate the concept of rotating a point clockwise around the origin. Suppose we have a point A with coordinates (3, 2). We want to rotate this point 180° clockwise around the origin.

Step 1: Identify the Original Coordinates

The original coordinates of point A are (3, 2).

Step 2: Apply the Rotation Formula

To rotate point A 180° clockwise around the origin, we use the rotation formula:

x' = -x y' = -y

Substituting the original coordinates (3, 2) into the formula, we get:

x' = -3 y' = -2

Step 3: Find the New Coordinates

The new coordinates of point A' after rotation are (-3, -2).

Conclusion

In this article, we explored the concept of rotating points in the coordinate plane and provided a step-by-step guide on how to find the coordinates of the rotated point. We used the rotation formula to rotate a point 180° clockwise around the origin and found the new coordinates of the rotated point. This concept has numerous applications in various fields, including geometry, trigonometry, and physics.

Frequently Asked Questions

Q: What is the rotation formula for rotating a point clockwise around the origin?

A: The rotation formula for rotating a point clockwise around the origin is:

x' = -x y' = -y

Q: How do I find the new coordinates of a rotated point?

A: To find the new coordinates of a rotated point, substitute the original coordinates into the rotation formula and simplify.

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

Additional Resources

For more information on rotating points in the coordinate plane, check out the following resources:

Discussion

Do you have any questions or comments about rotating points in the coordinate plane? Share your thoughts in the discussion section below!

Discussion Section

User 1: "I'm having trouble understanding the rotation formula. Can you provide more examples?"
User 2: "Yes, I'd love to see more examples. Can you also explain the difference between rotating a point clockwise and counterclockwise?"
User 3: "I'm trying to apply the rotation formula to a real-world problem. Can you provide more information on how to use this concept in a practical setting?"

Conclusion

Introduction

In our previous article, we explored the concept of rotating points in the coordinate plane and provided a step-by-step guide on how to find the coordinates of the rotated point. However, we know that there are many more questions and concerns that you may have about this topic. In this article, we will address some of the most frequently asked questions about rotating points in the coordinate plane.

Q: What is the rotation formula for rotating a point clockwise around the origin?

A: The rotation formula for rotating a point clockwise around the origin is:

x' = -x y' = -y

This formula is used to find the new coordinates of a point after it has been rotated 180° clockwise around the origin.

Q: How do I find the new coordinates of a rotated point?

A: To find the new coordinates of a rotated point, substitute the original coordinates into the rotation formula and simplify. For example, if you want to rotate the point (3, 2) 180° clockwise around the origin, you would use the following steps:

Substitute the original coordinates (3, 2) into the rotation formula: x' = -x y' = -y
Simplify the equation: x' = -3 y' = -2
The new coordinates of the rotated point are (-3, -2).

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

A: Rotating a point clockwise around the origin involves changing the sign of both the x and y coordinates, while rotating a point counterclockwise involves changing the sign of only one of the coordinates. For example, if you want to rotate the point (3, 2) 90° counterclockwise around the origin, you would use the following steps:

Substitute the original coordinates (3, 2) into the rotation formula: x' = y y' = -x
Simplify the equation: x' = 2 y' = -3
The new coordinates of the rotated point are (2, -3).

Q: Can I rotate a point more than 180°?

A: Yes, you can rotate a point more than 180°. However, you will need to use a different rotation formula. For example, if you want to rotate the point (3, 2) 270° clockwise around the origin, you would use the following steps:

Substitute the original coordinates (3, 2) into the rotation formula: x' = -y y' = x
Simplify the equation: x' = -2 y' = 3
The new coordinates of the rotated point are (-2, 3).

Q: Can I rotate a point by a fraction of a degree?

A: Yes, you can rotate a point by a fraction of a degree. However, you will need to use a different rotation formula. For example, if you want to rotate the point (3, 2) 45° clockwise around the origin, you would use the following steps:

Substitute the original coordinates (3, 2) into the rotation formula: x' = cos(45°)x - sin(45°)y y' = sin(45°)x + cos(45°)y
Simplify the equation: x' = 2.12 y' = 1.87
The new coordinates of the rotated point are (2.12, 1.87).

Q: Can I rotate a point in 3D space?

A: Yes, you can rotate a point in 3D space. However, you will need to use a different rotation formula. For example, if you want to rotate the point (3, 2, 1) 90° clockwise around the origin, you would use the following steps:

Substitute the original coordinates (3, 2, 1) into the rotation formula: x' = y y' = -x z' = z
Simplify the equation: x' = 2 y' = -3 z' = 1
The new coordinates of the rotated point are (2, -3, 1).

Conclusion

In conclusion, rotating points in the coordinate plane is a fundamental concept that has numerous applications in various fields. By understanding the rotation formula and how to apply it, you can solve problems involving rotations and gain a deeper understanding of the coordinate plane. We hope that this article has addressed some of the most frequently asked questions about rotating points in the coordinate plane and has provided you with a better understanding of this concept.

Additional Resources

For more information on rotating points in the coordinate plane, check out the following resources:

Discussion

Do you have any questions or comments about rotating points in the coordinate plane? Share your thoughts in the discussion section below!

Discussion Section

User 1: "I'm having trouble understanding the rotation formula. Can you provide more examples?"
User 2: "Yes, I'd love to see more examples. Can you also explain the difference between rotating a point clockwise and counterclockwise?"
User 3: "I'm trying to apply the rotation formula to a real-world problem. Can you provide more information on how to use this concept in a practical setting?"