To find out if a set of vectors is orthogonal, you must check a specific condition involving their dot products.
## Understanding Orthogonal Sets
An **orthogonal set of vectors** is a collection of vectors where every distinct pair of vectors is perpendicular to each other. In vector spaces, perpendicularity is defined using the dot product.
According to the definition:
Let {v₁, v₂,…,v<mark>k</mark>} be a subset of <mark>k</mark> distinct vectors of ℝ<sup><mark>n</mark></sup>. Then {v₁, v₂,…,v<mark>k</mark>} is an orthogonal set of vectors **if and only if the dot product of any two distinct vectors in this set is zero** — that is, if and only if v<mark>i</mark> · v<mark>j</mark> = 0, for 1 ≤ i, j ≤ <mark>k</mark>, i ≠ j.
This definition tells us exactly how to *identify* or *verify* if a given set meets the criteria for being orthogonal.
## The Method: Checking Dot Products
Based on the definition, the process to determine if a set of vectors {v₁, v₂,…,v<mark>k</mark>} is orthogonal is straightforward:
1. **Identify all distinct pairs** of vectors within the set. For a set with <mark>k</mark> vectors, these pairs are (v₁, v₂), (v₁, v₃), ..., (v₁, v<mark>k</mark>), (v₂, v₃), ..., (v<mark>k</mark>-₁, v<mark>k</mark>). Note that the order doesn't matter, so (v₁, v₂) is the same pair as (v₂, v₁), and we only consider pairs where the vectors are distinct (i ≠ j).
2. **Calculate the dot product** (also known as the scalar product) for each of these distinct pairs. The dot product of two vectors **u** = [u₁, u₂, ..., u<mark>n</mark>] and **v** = [v₁, v₂, ..., v<mark>n</mark>] in ℝ<sup><mark>n</mark></sup> is calculated as **u** · **v** = u₁v₁ + u₂v₂ + ... + u<mark>n</mark>v<mark>n</mark>.
3. **Check the result** for every dot product calculated.
* If the dot product of *every single distinct pair* of vectors in the set is zero (v<mark>i</mark> · v<mark>j</mark> = 0 for all i ≠ j), then the set is **orthogonal**.
* If *even one* dot product between a distinct pair of vectors is not zero, then the set is **not orthogonal**.
### Practical Steps
Here's a step-by-step guide:
* List the vectors in your set: {v₁, v₂, ..., v<mark>k</mark>}.
* Iterate through all unique pairs (v<mark>i</mark>, v<mark>j</mark>) where i < j.
* For each pair (v<mark>i</mark>, v<mark>j</mark>), compute v<mark>i</mark> · v<mark>j</mark>.
* If *all* these computed dot products are 0, conclude the set is orthogonal.
* If *any* computed dot product is non-zero, conclude the set is not orthogonal.
### Example
Let's check if the set of vectors S = {**v**₁, **v**₂, **v**₃} in ℝ³ is orthogonal, where **v**₁ = [1, 0, 0], **v**₂ = [0, 1, 0], and **v**₃ = [0, 0, 1].
The distinct pairs are (**v**₁, **v**₂), (**v**₁, **v**₃), and (**v**₂, **v**₃).
Let's calculate their dot products:
* **v**₁ · **v**₂ = (1)(0) + (0)(1) + (0)(0) = 0 + 0 + 0 = 0
* **v**₁ · **v**₃ = (1)(0) + (0)(0) + (0)(1) = 0 + 0 + 0 = 0
* **v**₂ · **v**₃ = (0)(0) + (1)(0) + (0)(1) = 0 + 0 + 0 = 0
| Pair | Dot Product Calculation | Result |
| :--------------- | :-------------------------------- | :----- |
| **v**₁, **v**₂ | (1)(0) + (0)(1) + (0)(0) | 0 |
| **v**₁, **v**₃ | (1)(0) + (0)(0) + (0)(1) | 0 |
| **v**₂, **v**₃ | (0)(0) + (1)(0) + (0)(1) | 0 |
Since the dot product of every distinct pair of vectors is zero, the set S = {**v**₁, **v**₂, **v**₃} *is* an orthogonal set.
### Why Orthogonality Matters
Orthogonal sets of non-zero vectors are always linearly independent. This property makes them particularly useful as bases for vector spaces, called orthogonal bases. They simplify many calculations in linear algebra, physics, and engineering.
