Understanding the Vector-Based Interpretation of the Pythagorean Theorem

While the traditional Pythagorean Theorem is a cornerstone of Euclidean geometry, it can be interpreted in the context of vector spaces. This article explores a vector-based proof where the area is represented as the vector for the Pythagorean Theorem. Although it may not be considered a 'rigorous proof' by some standards, it provides a fascinating perspective on the theorem's application in a broader mathematical context.

Introduction to Vector-Based Pythagorean Theorem

A Google search reveals that there are indeed vector-based proofs of the Pythagorean Theorem. These proofs utilize vectors to represent the sides of a right triangle, extending the theorem's applicability beyond Euclidean geometry. One such generalization involves the use of a symmetric bilinear form.

Generalized Pythagorean Theorem in Vector Spaces

The generalized Pythagorean Theorem can be applied in any vector space when distance and perpendicularity are defined via a symmetric bilinear form. Let's delve into the details of this concept.

Symmetric Bilinear Form and Quadrance

In a vector space, the dot product ( vec{u} cdot vec{v} ) is a symmetric bilinear form that maps two vectors to a scalar. The properties of this bilinear form include symmetry and linearity:

Symmetry: ( vec{u} cdot vec{v} vec{v} cdot vec{u} )

Linearity: ( alpha vec{u}_1 beta vec{u}_2 cdot vec{v} alpha vec{u}_1 cdot vec{v} beta vec{u}_2 cdot vec{v} )

Bilinearity: ( vec{u} cdot (alpha vec{v}_1 beta vec{v}_2) alpha vec{u} cdot vec{v}_1 beta vec{u} cdot vec{v}_2 )

For practical purposes, you can think of the usual Euclidean dot product. However, the theorem holds for any symmetric bilinear form. In this context, the square of the distance, known as quadrance, is defined as the self-dot product ( Qvec{v} vec{v} cdot vec{v} ).

Perpendicularity and the Generalized Pythagorean Theorem

Two vectors ( vec{u} ) and ( vec{v} ) are considered perpendicular if their dot product is zero: ( vec{u} perp vec{v} iff vec{u} cdot vec{v} 0 ).

Generalized Pythagorean Theorem

The generalized Pythagorean Theorem states that if two vectors are perpendicular, the sum of their quadrances equals the quadrance of their sum:

Theorem (Pythagoras): If ( vec{u} perp vec{v} ), then ( Qvec{u} Qvec{v} Q(vec{u} vec{v}) ).

Proof of the Generalized Pythagorean Theorem

Given vectors ( vec{u} ) and ( vec{v} ), we start by expressing the quadrance of their sum:

Step 1:

Expand ( Q(vec{u} vec{v}) ):

( Q(vec{u} vec{v}) (vec{u} vec{v}) cdot (vec{u} vec{v}) )

Step 2:

Apply the bilinearity and distributive properties of the dot product:

( Q(vec{u} vec{v}) (vec{u} vec{v}) cdot ( vec{u} vec{v} ) vec{u} cdot vec{u} vec{u} cdot vec{v} vec{v} cdot vec{u} vec{v} cdot vec{v} )

Step 3:

Use symmetry and linearity to simplify:

( Q(vec{u} vec{v}) vec{u} cdot vec{u} vec{v} cdot vec{v} vec{u} cdot vec{v} vec{v} cdot vec{u} vec{u} cdot vec{u} vec{v} cdot vec{v} 2 vec{u} cdot vec{v} )

Step 4:

Given that ( vec{u} perp vec{v} ), we have ( vec{u} cdot vec{v} 0 ) and ( vec{v} cdot vec{u} 0 ):

( Q(vec{u} vec{v}) vec{u} cdot vec{u} vec{v} cdot vec{v} Qvec{u} Qvec{v} )

Thus, the generalized Pythagorean Theorem is proven, and we can write:

Theorem (Pythagoras): If ( vec{u} perp vec{v} ), then ( Qvec{u} Qvec{v} Q(vec{u} vec{v}) ).

Conclusion

The vector-based interpretation of the Pythagorean Theorem opens up new avenues for understanding geometric relations in a broader context. This theorem not only extends the application of the Pythagorean Theorem but also provides a deeper insight into the mathematical structures underlying geometry.

Further Reading

For a more detailed exploration of these concepts, refer to the works of Norman Wildberger, particularly from his WildLinAlg series.