Question

By applying Pythagoras's theorem (the usual two-dimensional version) twice over, prove that the length \(r\) of a three-dimensional vector \(\mathbf{r}=(x, y, z)\) satisfies \(r^{2}=x^{2}+y^{2}+z^{2}\)

Short answer

\( r^2 = x^2 + y^2 + z^2 \) for a vector \( \mathbf{r} = (x, y, z) \).

Step-by-step solution

Consider the Base Case for Two Dimensions

In two dimensions, the position vector is \( \mathbf{r} = (x, y) \). According to Pythagoras's theorem, the length of this vector \( r \) is calculated as \( r_1 = \sqrt{x^2 + y^2} \). Therefore, \( r_1^2 = x^2 + y^2 \).

Incorporate the Third Dimension

Extend the two-dimensional space to three dimensions by adding the third component \( z \). The position vector then becomes \( \mathbf{r} = (x, y, z) \). Initially, ignore the \( z \) component and consider only the 2D calculation: \( r_1 = \sqrt{x^2 + y^2} \).

Apply Pythagoras's Theorem in Three Dimensions

Now, treat \( z \) as forming a right triangle with the hypotenuse \( r_1 \) and the length \( r \). Applying Pythagoras's theorem to this new triangle, we have: \( r = \sqrt{r_1^2 + z^2} \).

Substitute and Simplify

Substitute \( r_1^2 = x^2 + y^2 \) into the formula from Step 3: \( r = \sqrt{(x^2 + y^2) + z^2} \). Simplifying this expression, we get \( r = \sqrt{x^2 + y^2 + z^2} \). Consequently, \( r^2 = x^2 + y^2 + z^2 \).

Key concepts

Pythagorean theorem

In mathematics, the Pythagorean theorem is a fundamental principle used to determine the relationship between the sides of a right triangle. The theorem is best known for its application in two dimensions, where it states that the square of the hypotenuse's length is equal to the sum of the squares of the lengths of the other two sides. It is expressed as follows: \[ c^2 = a^2 + b^2 \] Here, \(c\) represents the hypotenuse, while \(a\) and \(b\) are the other two sides. This principle helps in calculating the distance in the Euclidean space, which can be extended to the third dimension.
To apply the Pythagorean theorem in three dimensions, we first handle two of the dimensions, calculating the hypotenuse for them, and then further apply the theorem including the third dimension. This process involves using the theorem twice, which is beautifully demonstrated through the exercise question. By applying the theorem twice, we derive the magnitude of a three-dimensional vector, connecting this ancient principle to modern-day vector mathematics.

three-dimensional vectors

A three-dimensional vector expands beyond the usual x-y plane to include depth, represented by the z-coordinate. Unlike two-dimensional vectors which lie flat, three-dimensional vectors give a more comprehensive picture of spatial positioning.
Mathematically, a three-dimensional vector is written as \( \mathbf{r} = (x, y, z) \), where \(x\), \(y\), and \(z\) are the magnitudes in their respective axes. Understanding these vectors is crucial in fields ranging from physics to computer graphics, where position, velocity, or forces are analyzed in three-dimensional space.
The exercise mentioned utilizes a three-dimensional vector, calculating its magnitude through a step-by-step enhancement of a two-dimensional approach. Using the concept of the Pythagorean theorem, we find the length of this vector, which incorporates all three spatial coordinates.

coordinate systems

Coordinate systems are essential in representing the particular location of points in space. In our daily applications, the most common coordinate system is the Cartesian system. This system uses perpendicular axes (x, y, and z) to define the position of any point in a two-dimensional or three-dimensional space.
The Cartesian coordinate system allows us to break down geometric problems, like computing the magnitude of vectors, with the help of coordinate measurements. Each coordinate represents an axis, which can be handled step-by-step as shown in the exercise solution.
Having a clear understanding of coordinate systems is essential, as it allows for consistent mathematical analysis and is pivotal in translating real-world objects to mathematical models. When dealing with three-dimensional spaces, knowing how these coordinate systems interact enables accurate calculations and designs. Consequently, strengthening our grasp on coordinate systems invariably improves our skill in tackling complex multi-dimensional problems.