Writing A Rule For A RotationAfter A Rotation, $A(-3,4$\] Maps To $A^{\prime}(4,3$\], $B(4,-5$\] Maps To $B^{\prime}(-5,-4$\], And $C(1,6$\] Maps To $C^{\prime}(6,-1$\]. Which Rule Describes The

Understanding the Concept of Rotation in Mathematics

Rotation is a fundamental concept in mathematics, particularly in geometry and trigonometry. It involves turning a figure or a point around a fixed point or axis by a certain angle. In this article, we will explore the concept of rotation and how to write a rule for a rotation.

Identifying the Type of Rotation

To write a rule for a rotation, we need to identify the type of rotation that has occurred. In this case, we are given three points: A(-3,4), B(4,-5), and C(1,6). After the rotation, these points map to A'(4,3), B'(-5,-4), and C'(6,-1). We need to determine the type of rotation that has occurred, which can be either a rotation about the origin or a rotation about a point other than the origin.

Rotation About the Origin

A rotation about the origin is a rotation that occurs around the point (0,0). In this case, the rotation matrix is given by:

$$\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$$

where $\theta$ is the angle of rotation.

Rotation About a Point Other Than the Origin

A rotation about a point other than the origin is a rotation that occurs around a point that is not the origin. In this case, the rotation matrix is given by:

$$\begin{bmatrix} \cos \theta & -\sin \theta & x \\ \sin \theta & \cos \theta & y \\ 0 & 0 & 1 \end{bmatrix}$$

where $(x,y)$ is the point around which the rotation occurs, and $\theta$ is the angle of rotation.

Writing a Rule for a Rotation

To write a rule for a rotation, we need to determine the type of rotation that has occurred and then use the rotation matrix to describe the rotation. In this case, we are given the points A(-3,4), B(4,-5), and C(1,6) and their images A'(4,3), B'(-5,-4), and C'(6,-1). We can use the rotation matrix to determine the angle of rotation and the point around which the rotation occurs.

Calculating the Angle of Rotation

To calculate the angle of rotation, we can use the following formula:

$$\tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1,y_1)$ is the original point and $(x_2,y_2)$ is the image of the point.

Calculating the Point Around Which the Rotation Occurs

To calculate the point around which the rotation occurs, we can use the following formula:

$$(x,y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

where $(x_1,y_1)$ is the original point and $(x_2,y_2)$ is the image of the point.

Applying the Rotation Matrix

Once we have determined the angle of rotation and the point around which the rotation occurs, we can apply the rotation matrix to describe the rotation. In this case, we can use the rotation matrix to describe the rotation of the points A(-3,4), B(4,-5), and C(1,6) to their images A'(4,3), B'(-5,-4), and C'(6,-1).

Conclusion

In conclusion, writing a rule for a rotation involves determining the type of rotation that has occurred and then using the rotation matrix to describe the rotation. We can use the rotation matrix to calculate the angle of rotation and the point around which the rotation occurs. By applying the rotation matrix, we can describe the rotation of points in a two-dimensional space.

Example

Let's consider an example to illustrate the concept of writing a rule for a rotation. Suppose we have a point A(2,3) and its image A'(5,4). We can use the rotation matrix to describe the rotation of the point A to its image A'.

Step 1: Determine the Type of Rotation

To determine the type of rotation, we need to calculate the angle of rotation and the point around which the rotation occurs. We can use the following formulas:

$$\tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$$

$$(x,y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

where $(x_1,y_1)$ is the original point and $(x_2,y_2)$ is the image of the point.

Step 2: Calculate the Angle of Rotation

Using the formula for the angle of rotation, we get:

$$\tan \theta = \frac{4 - 3}{5 - 2} = \frac{1}{3}$$

Step 3: Calculate the Point Around Which the Rotation Occurs

Using the formula for the point around which the rotation occurs, we get:

$$(x,y) = \left(\frac{2 + 5}{2}, \frac{3 + 4}{2}\right) = (3.5, 3.5)$$

Step 4: Apply the Rotation Matrix

Now that we have determined the angle of rotation and the point around which the rotation occurs, we can apply the rotation matrix to describe the rotation of the point A to its image A'.

Step 5: Write the Rule for the Rotation

The rule for the rotation is given by:

$$\begin{bmatrix} \cos \theta & -\sin \theta & x \\ \sin \theta & \cos \theta & y \\ 0 & 0 & 1 \end{bmatrix}$$

where $\theta$ is the angle of rotation, $(x,y)$ is the point around which the rotation occurs, and $(x_1,y_1)$ is the original point.

Final Answer

The final answer is: $\boxed{\begin{bmatrix} \cos \theta & -\sin \theta & 3.5 \\ \sin \theta & \cos \theta & 3.5 \\ 0 & 0 & 1 \end{bmatrix}}$

Frequently Asked Questions

Q: What is a rotation in mathematics?

A: A rotation is a transformation that turns a figure or a point around a fixed point or axis by a certain angle.

Q: What are the different types of rotations?

A: There are two main types of rotations: rotation about the origin and rotation about a point other than the origin.

Q: How do I determine the type of rotation that has occurred?

A: To determine the type of rotation, you need to calculate the angle of rotation and the point around which the rotation occurs. You can use the following formulas:

$$\tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$$

$$(x,y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

where $(x_1,y_1)$ is the original point and $(x_2,y_2)$ is the image of the point.

Q: How do I calculate the angle of rotation?

A: To calculate the angle of rotation, you can use the formula:

$$\tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$$

Q: How do I calculate the point around which the rotation occurs?

A: To calculate the point around which the rotation occurs, you can use the formula:

$$(x,y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Q: What is the rotation matrix?

A: The rotation matrix is a matrix that describes the rotation of a point or a figure. It is given by:

$$\begin{bmatrix} \cos \theta & -\sin \theta & x \\ \sin \theta & \cos \theta & y \\ 0 & 0 & 1 \end{bmatrix}$$

where $\theta$ is the angle of rotation, $(x,y)$ is the point around which the rotation occurs, and $(x_1,y_1)$ is the original point.

Q: How do I apply the rotation matrix?

A: To apply the rotation matrix, you need to multiply the matrix by the coordinates of the point or figure that you want to rotate.

Q: What is the final answer for the rotation matrix?

A: The final answer for the rotation matrix is:

$$\begin{bmatrix} \cos \theta & -\sin \theta & 3.5 \\ \sin \theta & \cos \theta & 3.5 \\ 0 & 0 & 1 \end{bmatrix}$$

Q: Can I use the rotation matrix to describe any type of rotation?

A: Yes, you can use the rotation matrix to describe any type of rotation, including rotation about the origin and rotation about a point other than the origin.

Q: How do I determine the angle of rotation for a rotation about a point other than the origin?

A: To determine the angle of rotation for a rotation about a point other than the origin, you need to calculate the angle of rotation using the formula:

$$\tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1,y_1)$ is the original point and $(x_2,y_2)$ is the image of the point.

Q: How do I determine the point around which the rotation occurs for a rotation about a point other than the origin?

A: To determine the point around which the rotation occurs for a rotation about a point other than the origin, you need to calculate the point using the formula:

$$(x,y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

where $(x_1,y_1)$ is the original point and $(x_2,y_2)$ is the image of the point.

Q: Can I use the rotation matrix to describe a rotation about a point other than the origin?

A: Yes, you can use the rotation matrix to describe a rotation about a point other than the origin. The rotation matrix is given by:

$$\begin{bmatrix} \cos \theta & -\sin \theta & x \\ \sin \theta & \cos \theta & y \\ 0 & 0 & 1 \end{bmatrix}$$

where $\theta$ is the angle of rotation, $(x,y)$ is the point around which the rotation occurs, and $(x_1,y_1)$ is the original point.

Q: How do I apply the rotation matrix for a rotation about a point other than the origin?

A: To apply the rotation matrix for a rotation about a point other than the origin, you need to multiply the matrix by the coordinates of the point or figure that you want to rotate.

Q: What is the final answer for the rotation matrix for a rotation about a point other than the origin?

A: The final answer for the rotation matrix for a rotation about a point other than the origin is:

$$\begin{bmatrix} \cos \theta & -\sin \theta & 3.5 \\ \sin \theta & \cos \theta & 3.5 \\ 0 & 0 & 1 \end{bmatrix}$$

Conclusion

In conclusion, writing a rule for a rotation involves determining the type of rotation that has occurred and then using the rotation matrix to describe the rotation. We can use the rotation matrix to calculate the angle of rotation and the point around which the rotation occurs. By applying the rotation matrix, we can describe the rotation of points in a two-dimensional space.