The Point V ( 4 , − 1 V (4, -1 V ( 4 , − 1 ] Is Rotated 90 ∘ 90^{\circ} 9 0 ∘ Clockwise Around The Origin. What Are The Coordinates Of Its Image V ′ V^{\prime} V ′ ?A. V ′ ( 1 , 4 V^{\prime}(1, 4 V ′ ( 1 , 4 ] B. V ′ ( − 1 , 4 V^{\prime}(-1, 4 V ′ ( − 1 , 4 ] C. V ′ ( − 1 , − 4 V^{\prime}(-1, -4 V ′ ( − 1 , − 4 ]

Introduction

In mathematics, rotation is a fundamental concept that plays a crucial role in various fields, including geometry, trigonometry, and physics. When a point is rotated around a fixed point, called the origin, its coordinates change in a predictable manner. In this article, we will explore the concept of point rotation and apply it to a specific problem: finding the coordinates of a point after a $90^{\circ}$ clockwise rotation around the origin.

Understanding Point Rotation

Point rotation is a transformation that changes the position of a point in a plane. When a point is rotated around the origin, its coordinates change according to the rotation angle and direction. There are two types of rotations: clockwise and counterclockwise. In this article, we will focus on clockwise rotation.

Clockwise Rotation Formula

The formula for a clockwise rotation of a point $(x, y)$ around the origin by an angle $\theta$ is given by:

$$x' = x \cos \theta + y \sin \theta$$

$$y' = -x \sin \theta + y \cos \theta$$

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

Applying the Formula to the Problem

In this problem, we are given a point $V(4, -1)$ that is rotated $90^{\circ}$ clockwise around the origin. We need to find the coordinates of its image $V'$. Using the formula above, we can substitute the values of $x$, $y$, and $\theta$ to find the new coordinates.

$$x' = 4 \cos 90^{\circ} + (-1) \sin 90^{\circ}$$

$$y' = -4 \sin 90^{\circ} + (-1) \cos 90^{\circ}$$

Since $\cos 90^{\circ} = 0$ and $\sin 90^{\circ} = 1$, we can simplify the equations:

$$x' = 4 \cdot 0 + (-1) \cdot 1 = -1$$

$$y' = -4 \cdot 1 + (-1) \cdot 0 = -4$$

Therefore, the coordinates of the image $V'$ are $(-1, -4)$.

Conclusion

In this article, we explored the concept of point rotation and applied it to a specific problem. We used the formula for clockwise rotation to find the coordinates of a point after a $90^{\circ}$ rotation around the origin. The result showed that the coordinates of the image $V'$ are $(-1, -4)$. This problem demonstrates the importance of understanding rotation in mathematics and its applications in various fields.

Discussion

The point rotation problem is a fundamental concept in mathematics that has numerous applications in various fields, including geometry, trigonometry, and physics. The formula for clockwise rotation provides a powerful tool for solving problems involving rotation. In this article, we applied the formula to a specific problem and found the coordinates of the image $V'$.

Answer

The correct answer is:

• C. $V^{\prime}(-1, -4)$

This answer is based on the calculation of the coordinates of the image $V'$ using the formula for clockwise rotation.

Additional Resources

For more information on point rotation and its applications, please refer to the following resources:

• Wikipedia: Rotation
• Math Open Reference: Rotation
• Khan Academy: Rotation

Q&A: Point Rotation and Its Applications

Q: What is point rotation?

A: Point rotation is a transformation that changes the position of a point in a plane. When a point is rotated around the origin, its coordinates change according to the rotation angle and direction.

Q: What are the two types of rotations?

A: There are two types of rotations: clockwise and counterclockwise. In this article, we focused on clockwise rotation.

Q: What is the formula for a clockwise rotation?

A: The formula for a clockwise rotation of a point $(x, y)$ around the origin by an angle $\theta$ is given by:

$$x' = x \cos \theta + y \sin \theta$$

$$y' = -x \sin \theta + y \cos \theta$$

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

Q: How do I apply the formula to a problem?

A: To apply the formula, substitute the values of $x$, $y$, and $\theta$ into the equations and simplify.

Q: What is the significance of the angle $\theta$ in the formula?

A: The angle $\theta$ determines the amount of rotation. A positive value of $\theta$ represents a clockwise rotation, while a negative value represents a counterclockwise rotation.

Q: Can I use the formula to find the coordinates of a point after a rotation?

A: Yes, the formula can be used to find the coordinates of a point after a rotation. Simply substitute the values of $x$, $y$, and $\theta$ into the equations and simplify.

Q: What are some real-world applications of point rotation?

A: Point rotation has numerous applications in various fields, including:

• Geometry and trigonometry
• Physics and engineering
• Computer graphics and animation
• Navigation and mapping

Q: How can I practice point rotation problems?

A: You can practice point rotation problems by using online resources, such as:

• Khan Academy: Rotation
• Math Open Reference: Rotation
• Wikipedia: Rotation

You can also try solving problems on your own using the formula and applying it to different scenarios.

Q: What are some common mistakes to avoid when working with point rotation?

A: Some common mistakes to avoid when working with point rotation include:

• Confusing the direction of rotation (clockwise vs. counterclockwise)
• Failing to account for the angle of rotation
• Not using the correct formula for the type of rotation

Q: Can I use point rotation to solve problems involving multiple rotations?

A: Yes, you can use point rotation to solve problems involving multiple rotations. Simply apply the formula for each rotation and combine the results.

Q: How can I use point rotation to solve problems involving reflections?

A: You can use point rotation to solve problems involving reflections by applying the formula for rotation and then reflecting the result across a line or axis.

Conclusion

Point rotation is a fundamental concept in mathematics that has numerous applications in various fields. By understanding the formula for clockwise rotation and how to apply it, you can solve problems involving point rotation and its applications. Remember to practice and review the material to ensure a solid understanding of the concept.

Additional Resources

For more information on point rotation and its applications, please refer to the following resources:

• Wikipedia: Rotation
• Math Open Reference: Rotation
• Khan Academy: Rotation

These resources provide a comprehensive overview of point rotation and its applications in mathematics and other fields.