What is the Difference Between Orthogonal and Orthonormal?
🆚 Go to Comparative Table 🆚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.
On this pageWhat is the Difference Between Orthogonal and Orthonormal? Comparative Table: Orthogonal vs Orthonormal
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:
Property | Orthogonal Vectors | Orthonormal Vectors |
---|---|---|
Definition | Two vectors are orthogonal if their inner product is zero. | Two vectors are orthonormal if they are orthogonal and have equal lengths. |
Length | The length of an orthogonal vector is not necessarily 1. | The length of an orthonormal vector is always 1. |
Inner Product | The inner product of two orthogonal vectors is zero. | The inner product of two orthonormal vectors is 1. |
Dot Product | The dot product of two orthogonal vectors is 0. | The dot product of two orthonormal vectors is 1. |
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.
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