A vector can be orthogonal to itself only if it is the zero vector.
In vector spaces, the concept of orthogonality between two vectors, say u and v (denoted as u ⊥ v), is formally defined using the inner product (often a dot product in common vector spaces like ℝ² or ℝ³). According to the provided reference:
Two vectors u,v∈V are orthogonal (denoted u⊥v) if ⟨u,v⟩=0. Note that the zero vector is the only vector that is orthogonal to itself. In fact, the zero vector is orthogonal to every vector v∈V.
This definition provides the key to understanding when a vector can be orthogonal to itself.
The Condition for Self-Orthogonality
For a vector v to be orthogonal to itself, it must satisfy the orthogonality condition where u = v. That is:
⟨v, v⟩ = 0
Let's explore this condition:
- The Inner Product: The inner product ⟨v, v⟩ represents the squared magnitude (or squared norm) of the vector v. In standard Euclidean space (like ℝⁿ), this is the sum of the squares of the vector's components.
- Zero Magnitude: The only vector whose magnitude is zero is the zero vector. If the squared magnitude ⟨v, v⟩ is zero, the magnitude ||v|| must also be zero. This is true if and only if v is the zero vector.
Therefore, the condition ⟨v, v⟩ = 0 is met exclusively by the zero vector.
Why Only the Zero Vector?
Consider a non-zero vector v. By definition, a non-zero vector has at least one component that is not zero. When you calculate the inner product of a non-zero vector with itself (its squared magnitude), you are summing the squares of its components. Since the square of any non-zero real number is positive, and the square of zero is zero, the sum of squares for a non-zero vector will always be a positive value, never zero.
For example, in ℝ²:
Let v = (x, y), where v ≠ (0, 0).
⟨v, v⟩ = x² + y²
If v is non-zero, either x ≠ 0 or y ≠ 0 (or both).
If x ≠ 0, then x² > 0.
If y ≠ 0, then y² > 0.
Thus, x² + y² > 0.
Now, consider the zero vector, 0. In any vector space, the zero vector has all its components equal to zero.
For example, in ℝ², 0 = (0, 0).
⟨0, 0⟩ = 0² + 0² = 0 + 0 = 0.
Since the inner product ⟨0, 0⟩ equals zero, the zero vector satisfies the condition for orthogonality to itself.
Summary: Key Points
- Orthogonality of a vector to itself means its inner product with itself is zero: ⟨v, v⟩ = 0.
- The inner product ⟨v, v⟩ represents the squared magnitude (||v||²) of the vector.
- The only vector with a squared magnitude of zero (and thus magnitude of zero) is the zero vector.
- Therefore, as stated in the reference, "the zero vector is the only vector that is orthogonal to itself."
This demonstrates that the zero vector holds a unique position in vector spaces, being the only vector that is orthogonal to itself and, notably, orthogonal to every other vector in the space.
