The Sequence Of Transformations, R O , 90 ∘ R X -axis R_{O, 90^\circ} R_{X\text{-axis}} R O , 9 0 ∘ ​ R X -axis ​ , Is Applied To Δ X Y Z \Delta XYZ Δ X Y Z To Produce Δ X ′ Y ′ Z ′ ′ \Delta X'Y'Z'' Δ X ′ Y ′ Z ′′ . If The Coordinates Of Y ′ Y' Y ′ Are ( 3 , 0 (3,0 ( 3 , 0 ], What Are The Coordinates Of Y Y Y ?

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Introduction

In geometry, transformations play a crucial role in understanding the properties and behavior of shapes. One of the fundamental transformations is the rotation of a point or a shape around a specific axis. In this article, we will explore the sequence of transformations, $R_{O, 90^\circ} r_{X\text{-axis}}$, applied to $\Delta XYZ$ to produce $\Delta X'Y'Z''$. We will focus on finding the coordinates of $Y$ given the coordinates of $Y'$.

Understanding the Sequence of Transformations

The sequence of transformations $R_{O, 90^\circ} r_{X\text{-axis}}$ involves two main steps:

1. Rotation around the origin: The first transformation is a rotation of $90^\circ$ around the origin $O$. This means that every point in the original shape will be rotated $90^\circ$ counterclockwise around the origin.
2. Reflection across the X-axis: The second transformation is a reflection of the rotated shape across the X-axis. This means that every point in the rotated shape will be reflected across the X-axis.

Applying the Sequence of Transformations to $\Delta XYZ$

Let's apply the sequence of transformations to $\Delta XYZ$ to produce $\Delta X'Y'Z''$. We will start with the original coordinates of $Y$, which we will denote as $(x, y)$.

Step 1: Rotation around the origin

When we rotate the point $Y$ around the origin by $90^\circ$, its new coordinates become $(-y, x)$. This is because the rotation matrix for a $90^\circ$ rotation around the origin is given by:

$$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$

Applying this rotation matrix to the coordinates of $Y$, we get:

$$\begin{bmatrix} -y \\ x \end{bmatrix}$$

Step 2: Reflection across the X-axis

When we reflect the rotated point $Y$ across the X-axis, its new coordinates become $(-y, -x)$. This is because the reflection matrix for a reflection across the X-axis is given by:

$$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

Applying this reflection matrix to the rotated coordinates of $Y$, we get:

$$\begin{bmatrix} -y \\ -x \end{bmatrix}$$

Finding the Coordinates of $Y$

We are given that the coordinates of $Y'$ are $(3, 0)$. We need to find the coordinates of $Y$, which we denoted as $(x, y)$. From the previous section, we know that the coordinates of $Y'$ are given by:

$$\begin{bmatrix} -y \\ -x \end{bmatrix}$$

Equating this to the given coordinates of $Y'$, we get:

$$\begin{bmatrix} -y \\ -x \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \end{bmatrix}$$

Solving for $x$ and $y$, we get:

$$-x = 3 \Rightarrow x = -3$$

$$-y = 0 \Rightarrow y = 0$$

Therefore, the coordinates of $Y$ are $(-3, 0)$.

Conclusion

In this article, we explored the sequence of transformations $R_{O, 90^\circ} r_{X\text{-axis}}$ applied to $\Delta XYZ$ to produce $\Delta X'Y'Z''$. We found the coordinates of $Y$ given the coordinates of $Y'$, which are $(3, 0)$. The coordinates of $Y$ are $(-3, 0)$.

References

• [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
• [2] "Linear Algebra and Its Applications" by Gilbert Strang

Further Reading

• [1] "Transformations in Geometry" by Michael Artin
• [2] "Geometry: A Modern Approach" by David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray
  

Introduction

In our previous article, we explored the sequence of transformations $R_{O, 90^\circ} r_{X\text{-axis}}$ applied to $\Delta XYZ$ to produce $\Delta X'Y'Z''$. We found the coordinates of $Y$ given the coordinates of $Y'$, which are $(3, 0)$. In this article, we will answer some frequently asked questions related to the sequence of transformations.

Q&A

Q: What is the purpose of the sequence of transformations?

A: The sequence of transformations $R_{O, 90^\circ} r_{X\text{-axis}}$ is used to rotate a point or a shape around the origin and then reflect it across the X-axis. This sequence of transformations is commonly used in geometry and trigonometry to solve problems involving rotations and reflections.

Q: What is the difference between a rotation and a reflection?

A: A rotation is a transformation that turns a point or a shape around a fixed point, called the center of rotation. A reflection, on the other hand, is a transformation that flips a point or a shape across a line or a plane.

Q: How do I apply the sequence of transformations to a point or a shape?

A: To apply the sequence of transformations to a point or a shape, you need to follow these steps:

1. Rotate the point or shape around the origin by $90^\circ$.
2. Reflect the rotated point or shape across the X-axis.

Q: What are the coordinates of $Y$ given the coordinates of $Y'$?

A: The coordinates of $Y$ are $(-3, 0)$ given the coordinates of $Y'$, which are $(3, 0)$.

Q: Can I apply the sequence of transformations to a shape with more than three vertices?

A: Yes, you can apply the sequence of transformations to a shape with more than three vertices. However, you need to apply the transformation to each vertex of the shape separately.

Q: How do I find the coordinates of a point after applying the sequence of transformations?

A: To find the coordinates of a point after applying the sequence of transformations, you need to follow these steps:

1. Rotate the point around the origin by $90^\circ$.
2. Reflect the rotated point across the X-axis.

Conclusion

In this article, we answered some frequently asked questions related to the sequence of transformations $R_{O, 90^\circ} r_{X\text{-axis}}$. We hope that this article has helped you to understand the sequence of transformations and how to apply it to points and shapes.

References

• [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
• [2] "Linear Algebra and Its Applications" by Gilbert Strang

Further Reading

• [1] "Transformations in Geometry" by Michael Artin
• [2] "Geometry: A Modern Approach" by David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray

Common Mistakes to Avoid

• Not applying the sequence of transformations correctly.
• Not understanding the difference between a rotation and a reflection.
• Not following the correct order of operations when applying the sequence of transformations.

Tips and Tricks

• Make sure to apply the sequence of transformations correctly.
• Use a diagram or a graph to visualize the transformation.
• Check your work by applying the inverse transformation to the transformed point or shape.