The Pythagorean theorem is a property of triangles first discovered in ancient Greece by the mathematician/philosopher Pythagoras. This theorem holds that in a right triangle (a triangle containing at least one angle equal to 90 degrees), the sum of the squares of the two shorter sides is equal to the square of the longest side, called the hypotenuse. This theorem has many applications in physics as it holds true for real objects and vectors.
The Pythagorean Theorem is often used in physics in vector addition. If you have two vectors at a 90-degree angle to each other, you can use the Pythagorean Theorem to find a sum vector's magnitude. For example, if a force of magnitude three is moving perpendicular to a vector of a magnitude four, the Pythagorean theorem will reveal that the sum of these vectors has a magnitude of five. Geometry or trigonometry is still necessary to find the new vector's angle, but this method provides the magnitude of the new angle.
Similarly, the Pythagorean Theorem can be used backwards to find a missing vector. If a physics problem gives the sum of two vectors and one of the vectors, the magnitude of the missing vector can be found with the Pythagorean Theorem. If you know that the hypotenuse is five, and one side of the triangle is three, you can algebraically rearrange the Pythagorean Theorem to find that the missing side is four.
Additionally, the Pythagorean Theorem can be used to find the X and Y components of initial velocity, useful in ballistics/projectile motion. In such an equation, the initial velocity is broken down into X and Y components. Trigonometry is used to find one component (cos of the angle times velocity for x, same equation with sin instead of cos for y). You can use both equations to find both vectors, or you can use one, then find the remaining component with the theorem.
A Word of Caution
Sometimes it appears the Pythagorean theorem works when it will not. A physics student should watch out for this. First, this method only works when adding two vectors. Do not use it to add more than two vectors. Also, this method only works when your angles form a right triangle. This means that your component vectors must be at a 90-degree angle to each other. Other methods work for vector addition, such as algebra, geometry or trigonometry in these situations.
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