Pythagorean theorem straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is
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