What is the Difference Between Orthogonal and Orthonormal?

The main difference between orthogonal and orthonormal vectors lies in their lengths. Both orthogonal and orthonormal vectors are perpendicular to each other, meaning their dot product is zero:

• Orthogonal vectors: These vectors have a dot product of zero, indicating that they are perpendicular to each other. For example, vectors $$u = [1, 2, 0]$$ and $$v = [0, 0, 3]$$ are orthogonal because $$u \cdot v = 1 \cdot 0 + 2 \cdot 0 + 0 \cdot 3 = 0$$.
• Orthonormal vectors: These vectors are not only perpendicular to each other but also have a length of 1. For example, vectors $$u = \left[\frac{1}{\sqrt{5}}, -\frac{1}{\sqrt{5}}, -\frac{1}{\sqrt{5}}\right]$$ and $$v = \left[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}\right]$$ are orthonormal because their dot product is zero, and both vectors have a length of 1.

In summary, orthonormal vectors are a special case of orthogonal vectors, where each vector has a length of 1. Two vectors are orthogonal if their dot product is zero, while they are orthonormal if their dot product is zero and they have a length of 1.

Comparative Table: Orthogonal vs Orthonormal

The main difference between orthogonal and orthonormal vectors lies in their lengths and inner products. Here is a table summarizing the differences:

In summary:

• Orthogonal vectors are perpendicular to each other, meaning they have a zero inner product.
• Orthonormal vectors are not only perpendicular to each other but also have equal lengths, making their inner products equal to 1.

Orthogonal vectors do not necessarily have equal lengths, whereas orthonormal vectors always have equal lengths and inner products of 1.