M(3, -6)$ Is Rotated $90^{\circ}$ Counterclockwise.Mapping Rule: $(x, Y) \rightarrow (-y, X)$1. Switch The $x$- And $y$-coordinates: $(6, -3)$2. Multiply The New

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Introduction

In coordinate geometry, rotations are an essential concept that helps us understand how points and shapes change when they are rotated around a fixed point or axis. In this article, we will explore the concept of rotating a point $(3, -6)$ by $90^{\circ}$ counterclockwise using the mapping rule $(x, y) \rightarrow (-y, x)$. We will break down the process into two steps and provide a detailed explanation of each step.

Step 1: Switching the x- and y-coordinates

The first step in rotating a point $(3, -6)$ by $90^{\circ}$ counterclockwise is to switch the x- and y-coordinates. This means that we will replace the x-coordinate with the y-coordinate and vice versa. Using the mapping rule $(x, y) \rightarrow (-y, x)$, we can write the new coordinates as $(y, x)$.

In this case, the new coordinates are $(6, -3)$. This is because the x-coordinate of the original point $(3, -6)$ is 3, and the y-coordinate is -6. When we switch the x- and y-coordinates, the new x-coordinate becomes -6, and the new y-coordinate becomes 3.

Step 2: Multiplying the new coordinates

The second step in rotating a point $(3, -6)$ by $90^{\circ}$ counterclockwise is to multiply the new coordinates by -1. This is because the mapping rule $(x, y) \rightarrow (-y, x)$ involves multiplying the new coordinates by -1.

In this case, the new coordinates are $(6, -3)$. When we multiply these coordinates by -1, we get $(6 \times -1, -3 \times -1)$, which simplifies to $(-6, 3)$.

Conclusion

In conclusion, rotating a point $(3, -6)$ by $90^{\circ}$ counterclockwise using the mapping rule $(x, y) \rightarrow (-y, x)$ involves two steps: switching the x- and y-coordinates and multiplying the new coordinates by -1. By following these steps, we can determine the new coordinates of the point after rotation.

Example Applications

Rotations are an essential concept in mathematics, and they have numerous applications in various fields, including:

  • Geometry: Rotations are used to study the properties of shapes and figures, such as circles, ellipses, and polygons.
  • Trigonometry: Rotations are used to solve problems involving right triangles and trigonometric functions.
  • Computer Graphics: Rotations are used to create 3D models and animations.
  • Engineering: Rotations are used to design and analyze mechanical systems, such as gears and linkages.

Tips and Tricks

Here are some tips and tricks to help you understand rotations:

  • Use a coordinate grid: Plotting points on a coordinate grid can help you visualize the rotation process.
  • Use a mapping rule: The mapping rule $(x, y) \rightarrow (-y, x)$ is a useful tool for rotating points.
  • Practice, practice, practice: The more you practice rotating points, the more comfortable you will become with the concept.

Common Mistakes

Here are some common mistakes to avoid when rotating points:

  • Confusing the x- and y-coordinates: Make sure to switch the x- and y-coordinates correctly.
  • Multiplying the wrong coordinates: Make sure to multiply the new coordinates by -1.
  • Not using a mapping rule: Using a mapping rule can help you avoid mistakes and ensure that you are rotating the point correctly.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about rotations in coordinate geometry.

Q: What is a rotation in coordinate geometry?

A: A rotation in coordinate geometry is a transformation that involves rotating a point or a shape around a fixed point or axis. This can be done by switching the x- and y-coordinates and multiplying the new coordinates by -1.

Q: What is the mapping rule for a 90° counterclockwise rotation?

A: The mapping rule for a 90° counterclockwise rotation is (x, y) → (-y, x). This means that we switch the x- and y-coordinates and multiply the new coordinates by -1.

Q: How do I rotate a point (3, -6) by 90° counterclockwise?

A: To rotate a point (3, -6) by 90° counterclockwise, we follow these steps:

  1. Switch the x- and y-coordinates: (6, -3)
  2. Multiply the new coordinates by -1: (-6, 3)

Q: What is the difference between a 90° counterclockwise rotation and a 90° clockwise rotation?

A: A 90° counterclockwise rotation involves switching the x- and y-coordinates and multiplying the new coordinates by -1, whereas a 90° clockwise rotation involves switching the x- and y-coordinates and multiplying the new coordinates by 1.

Q: Can I rotate a point by any angle?

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

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

A: To determine the new coordinates of a point after rotation, you can use the mapping rule for the specific angle of rotation.

Q: What are some common applications of rotations in coordinate geometry?

A: Rotations are used in various fields, including:

  • Geometry: Rotations are used to study the properties of shapes and figures, such as circles, ellipses, and polygons.
  • Trigonometry: Rotations are used to solve problems involving right triangles and trigonometric functions.
  • Computer Graphics: Rotations are used to create 3D models and animations.
  • Engineering: Rotations are used to design and analyze mechanical systems, such as gears and linkages.

Q: What are some tips and tricks for understanding rotations in coordinate geometry?

A: Here are some tips and tricks to help you understand rotations:

  • Use a coordinate grid to visualize the rotation process.
  • Use a mapping rule to ensure that you are rotating the point correctly.
  • Practice, practice, practice: The more you practice rotating points, the more comfortable you will become with the concept.

Q: What are some common mistakes to avoid when rotating points in coordinate geometry?

A: Here are some common mistakes to avoid when rotating points:

  • Confusing the x- and y-coordinates: Make sure to switch the x- and y-coordinates correctly.
  • Multiplying the wrong coordinates: Make sure to multiply the new coordinates by the correct value.
  • Not using a mapping rule: Using a mapping rule can help you avoid mistakes and ensure that you are rotating the point correctly.

Conclusion

In conclusion, rotations in coordinate geometry are an essential concept that involves rotating points and shapes around a fixed point or axis. By understanding the mapping rule and following the steps for rotation, you can determine the new coordinates of a point after rotation. Remember to practice, practice, practice to become comfortable with the concept of rotations.