What is the Pythagorean theorem of vectors?

The Pythagorean theorem of vectors is a fundamental principle stating that for any two orthogonal (perpendicular) vectors, the square of the magnitude of their difference is equal to the sum of the squares of their individual magnitudes.

Understanding the Vector Pythagorean Theorem

In Euclidean geometry, the Pythagorean theorem relates the sides of a right triangle: $a^2 + b^2 = c^2$, where a and b are the lengths of the legs and c is the length of the hypotenuse. In vector algebra, this theorem finds an analogous expression when considering orthogonal vectors.

Vectors can be thought of as directed line segments with both magnitude (length) and direction. When two vectors are orthogonal, they form a right angle.

The Pythagorean theorem for vectors, as stated in vector calculus, precisely defines this relationship:

Pythagoras theorem: if v and w are orthogonal, then |v − w|2 = |v|2 + |w|2.

Here:

• v and w represent vectors.
• |v| and |w| represent the magnitudes (lengths) of vectors v and w, respectively.
• |v|² and |w|² are the squares of their magnitudes.
• v − w is the vector resulting from subtracting vector w from vector v.
• |v − w| is the magnitude of the vector v − w.
• |v − w|² is the square of the magnitude of the vector v − w.

Geometrically, if you place vectors v and w tail-to-tail, and they are orthogonal, the vector v − w forms the third side of a right triangle. The magnitudes |v| and |w| correspond to the lengths of the two orthogonal legs, and the magnitude |v − w| corresponds to the length of the hypotenuse.

Condition for Orthogonality

The theorem strictly applies when the vectors are orthogonal. Two vectors v and w are orthogonal if and only if their dot product is zero (v ⋅ w = 0).

A special case mentioned in vector definitions is that the zero vector 0 is considered orthogonal to any vector. While perhaps counter-intuitive geometrically, this property holds algebraically (v ⋅ 0 = 0) and fits within the theorem's framework.

Practical Example

Let's consider the example vectors given in the reference: v = <2, 3> and w = <-3, 2>.

1. Check for Orthogonality: Calculate the dot product.
   v ⋅ w = (2 * -3) + (3 * 2) = -6 + 6 = 0
   Since the dot product is zero, v and w are indeed orthogonal.

2. Calculate Magnitudes Squared:
   |v|² = 2² + 3² = 4 + 9 = 13
|w|² = (-3)² + 2² = 9 + 4 = 13

3. Calculate Vector Difference and Magnitude Squared:
   v − w = <2 - (-3), 3 - 2> = <5, 1>
|v − w|² = 5² + 1² = 25 + 1 = 26

4. Verify the Theorem: Check if |v − w|² = |v|² + |w|².
   26 = 13 + 13
26 = 26
   The equation holds true, confirming the Pythagorean theorem for these orthogonal vectors.

Summary of Key Concepts

Concept	Description	Vector Notation
Vectors	Quantities with magnitude and direction	v, w
Orthogonal Vectors	Vectors that are perpendicular (dot product is 0)	v ⋅ w = 0
Magnitude	The length of a vector	|v|, |w|
Vector Difference	Result of subtracting one vector from another	v − w

The vector Pythagorean theorem is a specific instance of the geometric theorem, adapted for vector spaces, and it highlights the critical role of orthogonality in vector relationships.