Given A Point (-7, 4) On A Geometric Figure, What Is The New Point After Rotating The Figure 90 Degrees Counterclockwise?

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Introduction

Rotating a geometric figure involves turning it around a fixed point or axis by a certain angle. In this article, we will explore how to rotate a figure 90 degrees counterclockwise, given a point (-7, 4) on the figure. We will use the concept of coordinate geometry to find the new point after rotation.

Understanding Coordinate Geometry

Coordinate geometry is a branch of mathematics that deals with the study of points, lines, and shapes in a two-dimensional plane. Each point on the plane is represented by a pair of coordinates (x, y), where x is the horizontal distance from the origin (0, 0) and y is the vertical distance.

Rotating a Point 90 Degrees Counterclockwise

To rotate a point 90 degrees counterclockwise, we need to follow these steps:

  1. Identify the original point: The original point is given as (-7, 4).
  2. Determine the rotation axis: Since we are rotating 90 degrees counterclockwise, the rotation axis will be the y-axis.
  3. Apply the rotation formula: The rotation formula for a 90-degree counterclockwise rotation is:

(x', y') = (-y, x)

where (x', y') is the new point after rotation.

Applying the Rotation Formula

Now, let's apply the rotation formula to the original point (-7, 4).

x' = -y = -4 y' = x = -7

So, the new point after rotation is (-4, -7).

Visualizing the Rotation

To visualize the rotation, imagine a point on a coordinate plane. When you rotate it 90 degrees counterclockwise, the x-coordinate becomes the negative of the original y-coordinate, and the y-coordinate becomes the original x-coordinate.

Example Use Cases

Rotating a geometric figure 90 degrees counterclockwise has many practical applications in mathematics, physics, and engineering. Some examples include:

  • Computer graphics: Rotating objects in 3D space is a fundamental operation in computer graphics.
  • Physics: Rotating objects is essential in understanding the motion of objects in the physical world.
  • Engineering: Rotating parts and mechanisms is a critical aspect of engineering design.

Conclusion

In this article, we explored how to rotate a geometric figure 90 degrees counterclockwise, given a point (-7, 4) on the figure. We used the concept of coordinate geometry to find the new point after rotation. By following the rotation formula and visualizing the rotation, we can easily rotate a point 90 degrees counterclockwise.

Frequently Asked Questions

  • What is the new point after rotating a point 90 degrees counterclockwise? The new point is found by applying the rotation formula: (x', y') = (-y, x).
  • How do I visualize the rotation? Imagine a point on a coordinate plane and rotate it 90 degrees counterclockwise. The x-coordinate becomes the negative of the original y-coordinate, and the y-coordinate becomes the original x-coordinate.
  • What are some practical applications of rotating a geometric figure 90 degrees counterclockwise? Rotating objects in 3D space is essential in computer graphics, physics, and engineering.

Further Reading

  • Coordinate Geometry: A comprehensive guide to coordinate geometry, including points, lines, and shapes.
  • Rotation in 3D Space: A detailed explanation of rotating objects in 3D space, including the rotation matrix and Euler angles.
  • Computer Graphics: A comprehensive guide to computer graphics, including 3D modeling, rendering, and animation.

Introduction

Rotating a geometric figure 90 degrees counterclockwise is a fundamental concept in mathematics, physics, and engineering. In our previous article, we explored how to rotate a figure 90 degrees counterclockwise, given a point (-7, 4) on the figure. In this article, we will answer some frequently asked questions about rotating a geometric figure 90 degrees counterclockwise.

Q&A

Q1: What is the new point after rotating a point 90 degrees counterclockwise?

A1: The new point is found by applying the rotation formula: (x', y') = (-y, x).

Q2: How do I visualize the rotation?

A2: Imagine a point on a coordinate plane and rotate it 90 degrees counterclockwise. The x-coordinate becomes the negative of the original y-coordinate, and the y-coordinate becomes the original x-coordinate.

Q3: What are some practical applications of rotating a geometric figure 90 degrees counterclockwise?

A3: Rotating objects in 3D space is essential in computer graphics, physics, and engineering. Some examples include:

  • Computer graphics: Rotating objects in 3D space is a fundamental operation in computer graphics.
  • Physics: Rotating objects is essential in understanding the motion of objects in the physical world.
  • Engineering: Rotating parts and mechanisms is a critical aspect of engineering design.

Q4: How do I rotate a point 90 degrees clockwise?

A4: To rotate a point 90 degrees clockwise, you can use the rotation formula: (x', y') = (y, -x).

Q5: Can I rotate a point by any angle?

A5: Yes, you can rotate a point by any angle. However, the rotation formula will change depending on the angle of rotation.

Q6: How do I rotate a point in 3D space?

A6: Rotating a point in 3D space involves using the rotation matrix and Euler angles. The rotation matrix is a 3x3 matrix that represents the rotation of a point in 3D space.

Q7: What is the difference between rotating a point 90 degrees counterclockwise and rotating a point 90 degrees clockwise?

A7: The main difference between rotating a point 90 degrees counterclockwise and rotating a point 90 degrees clockwise is the direction of rotation. A counterclockwise rotation involves rotating the point in a counterclockwise direction, while a clockwise rotation involves rotating the point in a clockwise direction.

Q8: Can I rotate a point by a negative angle?

A8: Yes, you can rotate a point by a negative angle. However, the rotation formula will change depending on the sign of the angle.

Q9: How do I rotate a point by a multiple of 90 degrees?

A9: To rotate a point by a multiple of 90 degrees, you can use the rotation formula: (x', y') = (-y, x) for a 90-degree counterclockwise rotation, and (x', y') = (y, -x) for a 90-degree clockwise rotation.

Q10: What are some common mistakes to avoid when rotating a point?

A10: Some common mistakes to avoid when rotating a point include:

  • Incorrectly applying the rotation formula: Make sure to apply the correct rotation formula for the given angle of rotation.
  • Not considering the direction of rotation: Make sure to consider the direction of rotation when applying the rotation formula.
  • Not accounting for the sign of the angle: Make sure to account for the sign of the angle when applying the rotation formula.

Conclusion

In this article, we answered some frequently asked questions about rotating a geometric figure 90 degrees counterclockwise. We covered topics such as the new point after rotation, visualizing the rotation, practical applications, and common mistakes to avoid. By understanding these concepts, you can confidently rotate points in mathematics, physics, and engineering.

Further Reading

  • Coordinate Geometry: A comprehensive guide to coordinate geometry, including points, lines, and shapes.
  • Rotation in 3D Space: A detailed explanation of rotating objects in 3D space, including the rotation matrix and Euler angles.
  • Computer Graphics: A comprehensive guide to computer graphics, including 3D modeling, rendering, and animation.