Elementary algebra sign). One of the most well-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle: This equation states that formula_44, representing the square of the length of the side that is the hypotenuse (the side opposite the right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by formula_45 and formula_46. An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as formula_47); such equations are called
Pythagorean theorem the dot product of vectors. The dot product is called the "standard" inner product or the "Euclidean" inner product. However, other inner products are possible. The concept of length is replaced by the concept of the norm ||v|| of a vector v, defined as: In an inner-product space, the Pythagorean theorem states that for any two orthogonal vectors v and w we have Here the vectors v and w are akin to the sides of a right triangle with hypotenuse given by the vector sum v + w. This form of the Pythagorean theorem is a consequence of the properties