Problem 1 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

Let \((V, \cdot)\) be a real Euclidean inner product space and denote the length of a vector \(=\sqrt{x+x}\). Show that two vectors \(u\) and \(v\) are orthogonal iff \(|u+v|^{2}=|u|^{2}+|v|^{2} .\)

Step-by-Step Solution

Answer

Yes, two vectors \(u\) and \(v\) in the real Euclidean inner product space are orthogonal if and only if \(|u+v|^{2}=|u|^{2}+|v|^{2}\).

Step 1: Definition of Orthogonality

Start by remembering the definition of orthogonality. Two vectors \(u\) and \(v\) are orthogonal if their inner product is zero, i.e, \(u \cdot v = 0\).

Step 2: Define the Magnitude of vectors and their sum

Now, the magnitude or norm of a vector u in Euclidean space is described by \(|u| = \sqrt{u \cdot u}\), and similar for vector v. The magnitude of the vector sum is \(|u + v|^{2} = (u + v) \cdot (u + v)\) which can be expanded to \(u \cdot u + 2u \cdot v + v \cdot v = |u|^{2} + 2u \cdot v + |v|^{2}\).

Step 3: Compare and express the relation

Comparing the given equation \(|u + v|^{2}= |u|^{2} + |v|^{2}\) with the expression derived in step 2, it can be seen that the expressions are identical if \(u \cdot v = 0\), which, as per the definition of orthogonality, implies vectors u and v are orthogonal. Similarly, if u and v are orthogonal, then \(u \cdot v = 0\), which provides us the given equation.

Step 4: Conclude the proof

Hence, it can be concluded that two vectors \(u\) and \(v\) in the real Euclidean inner product space are orthogonal if and only if \(|u+v|^{2}=|u|^{2}+|v|^{2}\). This completes the proof.

Key Concepts

Vector Orthogonality

In a Euclidean inner product space, vector orthogonality refers to the condition where two vectors, say \(u\) and \(v\), are perpendicular to each other. This relationship is defined mathematically by the inner product, also called the dot product.

• If two vectors are orthogonal, their dot product equals zero, i.e., \(u \cdot v = 0\).
• The zero value indicates no projection of one vector onto the other, reflecting their perpendicularity.

Orthogonality is essential in vector mathematics, serving key roles in simplifying problems and proving various geometrical properties. Understanding this concept helps with recognizing patterns and symmetries within different vector spaces.

Magnitude of Vectors

The magnitude of a vector is a measure of its length. In a Euclidean inner product space, you calculate the magnitude by taking the square root of the dot product of the vector with itself.

• If \(u\) represents a vector, then its magnitude is given by \(|u| = \sqrt{u \cdot u}\).
• This formula provides a scalar value representing how long the vector extends in space.

Consider the sum of two vectors, \(u+v\). The magnitude of this sum can be expanded using their individual magnitudes and dot products. Hence, \[ |u + v|^{2} = (u + v) \cdot (u + v) = u \cdot u + 2u \cdot v + v \cdot v, \] showing how the magnitudes and interactions between \(u\) and \(v\) combine.

Orthogonal Vectors

Orthogonal vectors are vectors that meet the criterion of vector orthogonality — they are perpendicular in the Euclidean space, which means their dot product is zero.

• When vectors \(u\) and \(v\) are orthogonal, \(u \cdot v = 0\).
• This orthogonality alters the equation for the magnitude of their sum, transforming it to \(|u + v|^{2} = |u|^{2} + |v|^{2}\).

This characteristic means that when you calculate the magnitude of their sum, there's no contribution from the \(2u \cdot v\) term since it becomes zero. This simplification highlights a pivotal property of orthogonal vectors and aids in various mathematical proofs and applications like the Pythagorean theorem.