The Pythagorean Theorem in Physics

In physics, the Pythagorean theorem gives the size of a vector from its perpendicular parts. When two quantities act at right angles, such as 3 m/s east and 4 m/s north, their combined effect, called the resultant, has magnitude √3² + 4² = 5 m/s. The same rule gives vector lengths in 3D, projectile speeds and AC impedance.

Resultant vectors: adding perpendicular quantities

A vector is a quantity with both a size, called its magnitude, and a direction. Displacement, velocity, acceleration and force are vectors; mass and temperature are not. To add vectors you place them tip to tail. If the two vectors are perpendicular, they form the legs of a right triangle and the resultant is the hypotenuse:

Here R_x and R_y are the components, the parts of the vector along the x and y axes, and θ is the angle the resultant makes with the x-axis. The magnitude has the same unit as the components.

Breaking a vector into components

Physics problems often run the other way: you know a vector's magnitude and direction and need its components. Trigonometry does the split, and the theorem guarantees the pieces fit back together. A 50 N force pulling at 30° above the horizontal has

This always works because cos²θ + sin²θ = 1, the Pythagorean identity, which is the theorem written for a hypotenuse of length 1. Splitting, adding components and recombining with the theorem is the standard method for adding any number of vectors, the same bookkeeping navigators use for dead reckoning.

The magnitude of a vector in 3D

Real motion happens in three dimensions. Using the theorem twice, first across the floor and then up, gives the magnitude of a 3D vector:

The derivation is on the Pythagorean theorem in 3D page, and the pattern continues into any number of dimensions as the length of an n-component vector (see the theorem in n dimensions).

Projectile motion: speed from velocity components

A thrown ball has a horizontal velocity that stays constant (ignoring air resistance) and a vertical velocity that gravity changes by 9.8 m/s every second. Its speed at any moment is the hypotenuse of those two components.

Take a ball launched at 20 m/s, 30° above the horizontal, with g = 9.8 m/s²:

The theorem also explains why energy splits neatly into parts. Kinetic energy is 1 over 2mv², and v² = v_x² + v_y², so a 0.50 kg ball launched as above has 1 over 2(0.50)(20²) = 100 J of kinetic energy: 75 J from horizontal motion and 25 J from vertical motion. At the top, the vertical 25 J has all become potential energy, mgh = 0.50 × 9.8 × 5.1 ≈ 25 J, while the horizontal 75 J is untouched.

AC circuits: impedance Z = √(R² + X²)

In an alternating-current circuit, the voltage across a resistor rises and falls in step with the current, but the voltage across an inductor peaks a quarter cycle (90°) earlier and the voltage across a capacitor a quarter cycle later. Engineers draw these as arrows called phasors, and quantities 90° out of step combine exactly like perpendicular vectors.

The circuit's total opposition to current is its impedance Z, measured in ohms (Ω). The resistance R is one leg; the net reactance X = X_L − X_C is the other, where X_L = 2πfL for an inductor and X_C = 1 over 2πfC for a capacitor at frequency f:

The same right triangle appears for power. In Example 6, the real power is P = I²R = 172.8 W, the reactive power is Q = I²X = 230.4 var, and the apparent power is S = VI = 288 VA. Check: √172.8² + 230.4² = 288. Engineers call this the power triangle; it is the impedance triangle scaled by I².

Special relativity: the theorem with a minus sign

Einstein's theory of special relativity uses the ordinary Pythagorean theorem in its most famous thought experiment, and then produces a version of it with a twist. (Einstein also recalled finding his own proof of the theorem as a boy; see Einstein's proof.)

The light clock

Imagine a clock made of two mirrors a distance L apart, with a pulse of light bouncing between them. In the clock's own frame, one round trip takes Δτ = 2L/c, where c is the speed of light. Now watch the clock fly past at speed v. From the ground, the light travels along a diagonal while the mirrors move sideways, so each half-trip is the hypotenuse of a right triangle.

Apply the theorem to one half-trip, which takes time Δt/2 in the ground frame:

Multiply by 4, replace 2L with c Δτ, and solve for Δt:

This is time dilation. At v = 0.6c, √1 − 0.36 = 0.8, so each tick of the moving clock lasts 1 ÷ 0.8 = 1.25 times as long as seen from the ground.

The spacetime interval

Rearrange the same equation as (c Δτ)² = (c Δt)² − (v Δt)², where v Δt is the distance the clock moved. It says that a combination of time and distance comes out the same for every observer, even though they disagree about the time and the distance separately. In three space dimensions, this combination is the spacetime interval:

Compare it with ordinary distance, (Δx)² + (Δy)² + (Δz)², which stays the same however you rotate your axes. The spacetime version stays the same however fast you move, but the time part enters with the opposite sign. That single minus sign is the difference between the geometry of space and the geometry of spacetime. (Some textbooks flip every sign and write (Δx)² + (Δy)² + (Δz)² − (c Δt)²; the physics is identical.)

More places the theorem appears in physics

• Combining measurement uncertainties. Independent random errors combine as a root sum of squares: σ = √σ₁² + σ₂². A length with a 0.3 mm reading uncertainty and a 0.4 mm alignment uncertainty has a combined uncertainty of about 0.5 mm, not 0.7 mm.
• Energy and momentum. In relativity, a particle's energy E, momentum p and mass m satisfy E² = (pc)² + (mc²)². It has the form of the theorem, with rest energy mc² and pc as the legs and total energy as the hypotenuse; for light, m = 0 and E = pc.
• Distances between particles in simulations. Every gravity or collision calculation in a physics simulation starts with √(Δx)² + (Δy)² + (Δz)²; the programming page shows how code does it efficiently.