The Pythagorean theorem in 3D

The Pythagorean Theorem is a fundamental concept in mathematics that relates to the relationship between the sides of a right triangle. It states that the sum of the squares of the legs (the two sides that form the right angle) is equal to the square of the hypotenuse (the side opposite the right angle). This theorem can also be applied to 3D spaces, where right triangles exist.

In 3D space, a right triangle can be formed by two lines that intersect at a right angle and their connecting line, which is the hypotenuse. The Pythagorean Theorem can be applied to this triangle to find the length of the hypotenuse.

Let's consider a 3D right triangle with vertices at points A, B, and C. Suppose that AB and BC lie on the x-axis and y-axis, respectively, while the connecting line AC lies on the z-axis. The length of AB is represented by $a$, the length of BC is represented by $b$, and the length of AC is represented by $c$.

Using the Pythagorean Theorem, we have:

$c^2 = a^2 + b^2$

This equation is similar to the Pythagorean Theorem in 2D space, but the lengths now represent distances in 3D space.

Another way to think about this theorem in 3D is to use vectors. The vertices A, B, and C can be represented as position vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$, respectively. The sides of the triangle can then be represented as vectors:

$\vec{u} = \vec{b} - \vec{a}$

$\vec{v} = \vec{c} - \vec{a}$

The length of the hypotenuse can be found using the dot product of these vectors:

$\vec{u} \cdot \vec{v} = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \cos{\theta}$

where $\theta$ is the angle between vectors $\vec{u}$ and $\vec{v}$. Since the triangle is a right triangle, we know that $\theta = 90^{\circ}$, so $\cos{\theta} = 0$. This simplifies the equation to:

$\vec{u} \cdot \vec{v} = ||\vec{u}|| \cdot ||\vec{v}||$

Expanding the dot product yields:

$(\vec{b} - \vec{a}) \cdot (\vec{c} - \vec{a}) = ||\vec{b} - \vec{a}|| \cdot ||\vec{c} - \vec{a}||$

Squaring both sides gives:

$(\vec{b} - \vec{a}) \cdot (\vec{c} - \vec{a})^2 = ||\vec{b} - \vec{a}||^2 \cdot ||\vec{c} - \vec{a}||^2$

This is equivalent to the 3D version of the Pythagorean Theorem:

$(c - a)^2 + (b - a)^2 = c^2 + a^2 + b^2 - 2ab\cos{\theta}$

which simplifies to:

$c^2 = a^2 + b^2$

Therefore, the Pythagorean Theorem can be applied to 3D spaces using either distance or vector representations of the right triangle. It is a fundamental concept in mathematics and is useful in a variety of contexts, including physics and engineering.

Image Source: https://en.wikipedia.org/wiki/Pythagorean_theorem#Three_dimensions external_link

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3Dピタゴラスの定理[JA]