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Show a rotation traslation
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GPT-4o mini
To show a rotation and translation in a two-dimensional context, let's break down the concepts and provide an example.
### Rotation
A rotation involves turning a point or a shape around a specific point, called the center of rotation. For example, if we rotate a point \((x, y)\) around the origin \((0, 0)\) by an angle \(\theta\), the new coordinates \((x', y')\) can be calculated using the following formulas:
- \(x' = x \cdot \cos(\theta) - y \cdot \sin(\theta)\)
- \(y' = x \cdot \sin(\theta) + y \cdot \cos(\theta)\)
### Translation
Translation involves moving a point or a shape from one location to another without rotating or resizing it. If we translate a point \((x, y)\) by a vector \((a, b)\), the new coordinates \((x', y')\) will be:
- \(x' = x + a\)
- \(y' = y + b\)
### Example
Let’s say we have a point \(P(1, 2)\) that we want to rotate 90 degrees counterclockwise about the origin, and then translate it by the vector \((3, 4)\).
1. **Rotation of \(P(1, 2)\) by \(\theta = 90^\circ\):**
- Convert degrees to radians: \(90^\circ = \frac{\pi}{2} \, \text{radians}\)
- Calculate the new coordinates:
- \(x' = 1 \cdot \cos\left(\frac{\pi}{2}\right) - 2 \cdot \sin\left(\frac{\pi}{2}\right) = 0 - 2 = -2\)
- \(y' = 1 \cdot \sin\left(\frac{\pi}{2}\right) + 2 \cdot \cos\left(\frac{\pi}{2}\right) = 1 + 0 = 1\)
- So after rotation, the new point \(P' = (-2, 1)\).
2. **Translation of \(P'(-2, 1)\) by \((3, 4)\):**
- Calculate the new coordinates:
- \(x' = -2 + 3 = 1\)
- \(y' = 1 + 4 = 5\)
- So after translation, the final point is \(P'' = (1, 5)\).
### Summary
Starting from point \(P(1, 2)\):
1. Rotating \(P(1, 2)\) by 90 degrees gives \(P'(-2, 1)\).
2. Translating \(P'(-2, 1)\) by vector \((3, 4)\) gives \(P''(1, 5)\).
This is a basic example of how rotation and translation can be applied to a point in two-dimensional space!
### Rotation
A rotation involves turning a point or a shape around a specific point, called the center of rotation. For example, if we rotate a point \((x, y)\) around the origin \((0, 0)\) by an angle \(\theta\), the new coordinates \((x', y')\) can be calculated using the following formulas:
- \(x' = x \cdot \cos(\theta) - y \cdot \sin(\theta)\)
- \(y' = x \cdot \sin(\theta) + y \cdot \cos(\theta)\)
### Translation
Translation involves moving a point or a shape from one location to another without rotating or resizing it. If we translate a point \((x, y)\) by a vector \((a, b)\), the new coordinates \((x', y')\) will be:
- \(x' = x + a\)
- \(y' = y + b\)
### Example
Let’s say we have a point \(P(1, 2)\) that we want to rotate 90 degrees counterclockwise about the origin, and then translate it by the vector \((3, 4)\).
1. **Rotation of \(P(1, 2)\) by \(\theta = 90^\circ\):**
- Convert degrees to radians: \(90^\circ = \frac{\pi}{2} \, \text{radians}\)
- Calculate the new coordinates:
- \(x' = 1 \cdot \cos\left(\frac{\pi}{2}\right) - 2 \cdot \sin\left(\frac{\pi}{2}\right) = 0 - 2 = -2\)
- \(y' = 1 \cdot \sin\left(\frac{\pi}{2}\right) + 2 \cdot \cos\left(\frac{\pi}{2}\right) = 1 + 0 = 1\)
- So after rotation, the new point \(P' = (-2, 1)\).
2. **Translation of \(P'(-2, 1)\) by \((3, 4)\):**
- Calculate the new coordinates:
- \(x' = -2 + 3 = 1\)
- \(y' = 1 + 4 = 5\)
- So after translation, the final point is \(P'' = (1, 5)\).
### Summary
Starting from point \(P(1, 2)\):
1. Rotating \(P(1, 2)\) by 90 degrees gives \(P'(-2, 1)\).
2. Translating \(P'(-2, 1)\) by vector \((3, 4)\) gives \(P''(1, 5)\).
This is a basic example of how rotation and translation can be applied to a point in two-dimensional space!