Here’s a proof of the Pythagorean Theorem by way of vectors. Of course, if your students already know vectors, they’re already way past the Pythagorean Theorem, but I thought Richard Pennington‘s statement of this on LinkedIn gave a pretty and stunningly brief (after all the definitions) proof of one of mathematics’ greatest equations.
Let O be the origin, and let and be two position vectors starting at O. The vector from to is simply , which I will call . Using properties of dot products,
The dot product of a vector with itself is the square of its magnitude, so
where is the angle between and .
This is the Law of Cosines–in my classes, I call it the generalized Pythagorean Theorem for all triangles. If , then and are the legs of a right triangle with hypotenuse which makes and