Understanding Rotation in Coordinate Geometry

Oct 13, 2025 by magicalceylontours.com 55 views

The Point ${$ V (4, -1) $}$ Rotated ${$ 90^{\circ} $}$ Clockwise Around the Origin

Understanding Rotation in Coordinate Geometry

Rotation is a fundamental concept in coordinate geometry, where a point or a shape is turned around a fixed point called the origin. In this article, we will explore the concept of rotation and how it affects the coordinates of a point. We will specifically focus on rotating a point ${$ V (4, -1) $}$ by ${$ 90^{\circ} $}$ clockwise around the origin.

What is Rotation?

Rotation is a transformation that turns a point or a shape around a fixed point, called the origin. The origin is the point (0, 0) in the coordinate plane. When a point is rotated around the origin, its distance from the origin remains the same, but its position changes. The new position of the point is determined by the angle of rotation and the initial position of the point.

Types of Rotation

There are two types of rotation: clockwise and counterclockwise. Clockwise rotation is a rotation in the direction of the hands of a clock, while counterclockwise rotation is a rotation in the opposite direction. In this article, we will focus on clockwise rotation.

Rotating a Point Around the Origin

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

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

where ${$ (x, y) $}$ are the initial coordinates of the point, ${$ (x', y') $}$ are the new coordinates of the point after rotation, and ${$ \theta $}$ is the angle of rotation.

Rotating the Point ${$ V (4, -1) $}$ by ${$ 90^{\circ} $}$ Clockwise

To rotate the point ${$ V (4, -1) $}$ by ${$ 90^{\circ} $}$ clockwise, we can use the formula above. We have ${$ x = 4 $}$, ${$ y = -1 $}$, and ${$ \theta = 90^{\circ} $}$.

Substituting these values into the formula, we get:

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

Using the values of ${$ \cos 90^{\circ} = 0 $}$ and ${$ \sin 90^{\circ} = 1 $}$, we get:

${$ x' = 4 (0) - (-1) (1) $}{$ x' = 1 $}$

${$ y' = 4 (1) + (-1) (0) $}{$ y' = 4 $}$

Therefore, the new coordinates of the point ${$ V $}$ after rotation are ${$ (1, 4) $}$.

Conclusion

In this article, we explored the concept of rotation in coordinate geometry and how it affects the coordinates of a point. We specifically focused on rotating a point ${$ V (4, -1) $}$ by ${$ 90^{\circ} $}$ clockwise around the origin. We used the formula for rotation and calculated the new coordinates of the point after rotation. The new coordinates of the point ${$ V $}$ after rotation are ${$ (1, 4) $}$.

Answer

The coordinates of the image ${$ V^{\prime} $}$ of the point ${$ V (4, -1) $}$ rotated ${$ 90^{\circ} $}$ clockwise around the origin are ${$ V^{\prime}(1, 4) $}$.

Discussion

The concept of rotation is an important one in coordinate geometry, and it has many applications in mathematics and science. In this article, we focused on rotating a point around the origin, but rotation can also be applied to shapes and other geometric objects. The formula for rotation can be used to rotate any point or shape around the origin, and it is a powerful tool for solving problems in mathematics and science.

Related Topics

Rotation in coordinate geometry
Transformation of points and shapes
Angle of rotation
Formula for rotation
Coordinate geometry
Mathematics
Science

References

[1] "Coordinate Geometry" by Khan Academy
[2] "Rotation in Coordinate Geometry" by Math Open Reference
[3] "Transformation of Points and Shapes" by Wolfram MathWorld
Q&A: Rotating Points and Shapes in Coordinate Geometry

Understanding Rotation in Coordinate Geometry

Rotation is a fundamental concept in coordinate geometry, where a point or a shape is turned around a fixed point called the origin. In this article, we will explore the concept of rotation and how it affects the coordinates of a point. We will also answer some frequently asked questions about rotation in coordinate geometry.

Q: What is rotation in coordinate geometry?

A: Rotation is a transformation that turns a point or a shape around a fixed point, called the origin. The origin is the point (0, 0) in the coordinate plane. When a point is rotated around the origin, its distance from the origin remains the same, but its position changes.

Q: What are the two types of rotation?

A: There are two types of rotation: clockwise and counterclockwise. Clockwise rotation is a rotation in the direction of the hands of a clock, while counterclockwise rotation is a rotation in the opposite direction.

Q: How do I rotate a point around the origin?

A: To rotate a point around the origin, you can use the following formula:

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

where ${$ (x, y) $}$ are the initial coordinates of the point, ${$ (x', y') $}$ are the new coordinates of the point after rotation, and ${$ \theta $}$ is the angle of rotation.

Q: What is the formula for rotation?

A: The formula for rotation is:

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

Q: How do I calculate the new coordinates of a point after rotation?

A: To calculate the new coordinates of a point after rotation, you can use the formula above. Simply substitute the initial coordinates of the point and the angle of rotation into the formula, and solve for the new coordinates.

Q: What is the angle of rotation?

A: The angle of rotation is the amount of rotation in degrees. For example, a rotation of ${$ 90^{\circ} $}$ is a rotation of 90 degrees.

Q: How do I determine the direction of rotation?

A: The direction of rotation is determined by the sign of the angle of rotation. A positive angle of rotation indicates a clockwise rotation, while a negative angle of rotation indicates a counterclockwise rotation.

Q: Can I rotate a shape around the origin?

A: Yes, you can rotate a shape around the origin. The formula for rotation can be used to rotate any point or shape around the origin.

Q: What are some real-world applications of rotation in coordinate geometry?

A: Rotation has many real-world applications in mathematics and science. Some examples include:

Navigation: Rotation is used in navigation to determine the direction of travel.
Engineering: Rotation is used in engineering to design and build machines and mechanisms.
Computer graphics: Rotation is used in computer graphics to create 3D models and animations.

Conclusion

In this article, we explored the concept of rotation in coordinate geometry and answered some frequently asked questions about rotation. We also discussed the formula for rotation and how to calculate the new coordinates of a point after rotation. Rotation is a fundamental concept in coordinate geometry, and it has many real-world applications in mathematics and science.

Related Topics

Rotation in coordinate geometry
Transformation of points and shapes
Angle of rotation
Formula for rotation
Coordinate geometry
Mathematics
Science

References

[1] "Coordinate Geometry" by Khan Academy
[2] "Rotation in Coordinate Geometry" by Math Open Reference
[3] "Transformation of Points and Shapes" by Wolfram MathWorld