Rotation Around a Fixed Point

Oct 13, 2025 by miodeya.com 132 views

In mathematics, a rotation is a fundamental transformation that involves rotating a point or a shape around a fixed point or axis. This transformation is commonly used in geometry, trigonometry, and other branches of mathematics. In this article, we will explore the concept of rotation and how it can be described using a specific rule.

What is a Rotation?

A rotation is a transformation that involves rotating a point or a shape around a fixed point or axis. This transformation can be thought of as a movement of the point or shape in a circular motion around the fixed point or axis. The amount of rotation can be measured in degrees, and it can be clockwise or counterclockwise.

Types of Rotations

There are two main types of rotations: rotations around a fixed point and rotations around a fixed axis. A rotation around a fixed point involves rotating a point or a shape around a fixed point, while a rotation around a fixed axis involves rotating a point or a shape around a fixed axis.

Rotation Around a Fixed Point

A rotation around a fixed point involves rotating a point or a shape around a fixed point. This type of rotation can be described using a specific rule, which involves multiplying the coordinates of the point by a rotation matrix.

Rotation Around a Fixed Axis

A rotation around a fixed axis involves rotating a point or a shape around a fixed axis. This type of rotation can be described using a specific rule, which involves multiplying the coordinates of the point by a rotation matrix.

The Rotation Rule

The rotation rule is a mathematical formula that describes how a point or a shape is rotated around a fixed point or axis. This rule involves multiplying the coordinates of the point by a rotation matrix, which is a square matrix that represents the rotation.

Example: Rotation of Points

Let's consider an example where we have three points: A(-3,4), B(4,-5), and C(1,6). We are given that A maps to A'(4,3), B maps to B'(-5,-4), and C maps to C'(6,-1). We need to find the rotation rule that describes this transformation.

Step 1: Find the Rotation Matrix

To find the rotation matrix, we need to determine the angle of rotation and the center of rotation. Let's assume that the center of rotation is the origin (0,0). We can find the angle of rotation by using the coordinates of the points.

Step 2: Determine the Angle of Rotation

We can determine the angle of rotation by using the coordinates of the points. Let's consider the point A(-3,4) and its image A'(4,3). We can find the angle of rotation by using the following formula:

θ = arctan((y2 - y1) / (x2 - x1))

where (x1, y1) are the coordinates of the point A and (x2, y2) are the coordinates of the point A'.

Step 3: Find the Rotation Rule

Once we have determined the angle of rotation, we can find the rotation rule by multiplying the coordinates of the points by a rotation matrix. The rotation matrix is a square matrix that represents the rotation.

Step 4: Verify the Rotation Rule

Once we have found the rotation rule, we can verify it by applying it to the points and checking if the resulting points match the given images.

Conclusion

In this article, we have explored the concept of rotation and how it can be described using a specific rule. We have seen how to find the rotation matrix, determine the angle of rotation, and find the rotation rule. We have also seen how to verify the rotation rule by applying it to the points and checking if the resulting points match the given images.

Rotation Rule

Rotation Matrix

A rotation matrix is a square matrix that represents the rotation. It is used to multiply the coordinates of the point to find the resulting point after the rotation.

Rotation Formula

The rotation formula is a mathematical formula that describes how a point or a shape is rotated around a fixed point or axis. This formula involves multiplying the coordinates of the point by a rotation matrix.

Rotation Example

Step 1: Find the Rotation Matrix

Step 2: Determine the Angle of Rotation

θ = arctan((y2 - y1) / (x2 - x1))

where (x1, y1) are the coordinates of the point A and (x2, y2) are the coordinates of the point A'.

Step 3: Find the Rotation Rule

Step 4: Verify the Rotation Rule

Once we have found the rotation rule, we can verify it by applying it to the points and checking if the resulting points match the given images.

Rotation Rule Formula

The rotation rule formula is a mathematical formula that describes how a point or a shape is rotated around a fixed point or axis. This formula involves multiplying the coordinates of the point by a rotation matrix.

Rotation Matrix Formula

The rotation matrix formula is a mathematical formula that represents the rotation. It is used to multiply the coordinates of the point to find the resulting point after the rotation.

Conclusion

Rotation Rule Summary

Rotation Matrix Summary

A rotation matrix is a square matrix that represents the rotation. It is used to multiply the coordinates of the point to find the resulting point after the rotation.

Rotation Formula Summary

Rotation Example Summary

Step 1: Find the Rotation Matrix

Step 2: Determine the Angle of Rotation

θ = arctan((y2 - y1) / (x2 - x1))

where (x1, y1) are the coordinates of the point A and (x2, y2) are the coordinates of the point A'.

Step 3: Find the Rotation Rule

Step 4: Verify the Rotation Rule

Once we have found the rotation rule, we can verify it by applying it to the points and checking if the resulting points match the given images.

Rotation Rule Formula

Rotation Matrix Formula

The rotation matrix formula is a mathematical formula that represents the rotation. It is used to multiply the coordinates of the point to find the resulting point after the rotation.

Conclusion

In this article, we have explored the concept of rotation and how it can be described using a specific rule. We have seen how to find the rotation matrix