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Pythagorean theorem in three demensions preview
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GPT-4o mini
The Pythagorean theorem traditionally applies to two-dimensional space, stating that in a right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)):
\[
c^2 = a^2 + b^2
\]
In three-dimensional space, the Pythagorean theorem can be extended to determine the distance between two points. If you have two points in 3D space represented as \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\), the distance \(d\) between these two points can be calculated using a generalized form of the Pythagorean theorem:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
This formula derives from the concept of forming a right triangle in three dimensions. To visualize this, consider the following steps:
1. **Project the Points onto the Coordinates**:
- First, project point \(A\) and point \(B\) onto the \(xy\) plane, where the distance between the two projected points (\(A'\) and \(B'\)) can be calculated using the 2D Pythagorean theorem:
\[
d_{xy} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
2. **Consider the Height (z-axis)**:
- Next, consider the vertical distance in the \(z\) direction:
\[
d_z = |z_2 - z_1|
\]
3. **Combine the Distances**:
- Finally, combine these two distances in 3D using the Pythagorean theorem:
\[
d = \sqrt{d_{xy}^2 + d_z^2} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
This extension of the Pythagorean theorem to three dimensions allows us to calculate distances in three-dimensional space effectively, making it a foundational concept in geometry and various applications in physics, engineering, and computer science.
\[
c^2 = a^2 + b^2
\]
In three-dimensional space, the Pythagorean theorem can be extended to determine the distance between two points. If you have two points in 3D space represented as \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\), the distance \(d\) between these two points can be calculated using a generalized form of the Pythagorean theorem:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
This formula derives from the concept of forming a right triangle in three dimensions. To visualize this, consider the following steps:
1. **Project the Points onto the Coordinates**:
- First, project point \(A\) and point \(B\) onto the \(xy\) plane, where the distance between the two projected points (\(A'\) and \(B'\)) can be calculated using the 2D Pythagorean theorem:
\[
d_{xy} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
2. **Consider the Height (z-axis)**:
- Next, consider the vertical distance in the \(z\) direction:
\[
d_z = |z_2 - z_1|
\]
3. **Combine the Distances**:
- Finally, combine these two distances in 3D using the Pythagorean theorem:
\[
d = \sqrt{d_{xy}^2 + d_z^2} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
This extension of the Pythagorean theorem to three dimensions allows us to calculate distances in three-dimensional space effectively, making it a foundational concept in geometry and various applications in physics, engineering, and computer science.