# Matrices



# New Page

**✅ **Vectors describe positions or directions** within a coordinate space.  
✅ **Matrices describe how to transform vectors** from one space to another.**

> \[ Xaxis.x Yaxis.x Zaxis.x Tx \]  
> \[ Xaxis.y Yaxis.y Zaxis.y Ty \]  
> \[ Xaxis.z Yaxis.z Zaxis.z Tz \]  
> \[ 0 0 0 1 \]
> 
> A 4×4 transformation matrix is made up of:

- **3 direction vectors** (X, Y, Z axes — orientation)
- **1 position vector** (T — translation)
- **1 extra row** that supports **homogeneous coordinates**

<table data-end="873" data-start="338" id="bkmrk-action-result-set-w-"><thead data-end="423" data-start="338"><tr data-end="423" data-start="338"><th data-end="364" data-start="338">Action</th><th data-end="423" data-start="364">Result</th></tr></thead><tbody data-end="873" data-start="510"><tr data-end="610" data-start="510"><td>Set `w = 1`</td><td>Vector becomes a **position** → gets full matrix transform (T \* R \* S)</td></tr><tr data-end="696" data-start="611"><td>Set `w = 0`</td><td>Vector becomes a **direction** → skips translation</td></tr><tr data-end="782" data-start="697"><td>Divide by `w`</td><td>Converts **clip space → NDC** (perspective divide)</td></tr><tr data-end="873" data-start="783"><td>Store something in `w`</td><td>Use `w` as a **packed value** (for instancing, motion, etc.)</td></tr></tbody></table>

YES — asking *what is a basis* means you’re ready to **understand all of 3D spacebuilding and matrix logic**. Let’s break it down clearly, intuitively, and practically for tech art 👇

---

## What Is a **Basis**?

> A **basis** is a set of vectors that define a coordinate space.  
> It tells you:

- Where the **origin** is
- What direction **X, Y, Z** go
- What **"1 unit"** means in each direction

It’s like building a mini coordinate system — your own local grid.

---

### ✅ Example: Standard Basis in 3D

The default basis in world space is:

```plaintext
X = (1, 0, 0)
Y = (0, 1, 0)
Z = (0, 0, 1)

```

This is called an **orthonormal basis**:

- Each axis is **unit length**
- Each axis is **perpendicular** to the others

This is what Unity uses for world space, object space (by default), and more.

---

## 🧩 Why Is a Basis Useful?

> A basis lets you **build your own space** inside another space.

You can:

- Rotate it (to align to a surface)
- Scale it (to stretch/compress space)
- Offset it (with a position = origin)
- Pack it into a matrix → for transforming vectors

---

### 🧱 Basis = The Building Blocks of a Matrix

In a transformation matrix, the 3 main columns **are your basis vectors**:

```plaintext
[ Right  Up  Forward  Position ]
[   X     Y     Z        T     ]

```

So:

- The **first 3 columns** = your basis vectors (local X, Y, Z)
- The **4th column** = your new origin (translation vector)

✅ A matrix is just a **basis with a location**.

---

## 🧪 Real Tech Art Examples

<table id="bkmrk-use-case-basis-invol"><thead><tr><th>Use Case</th><th>Basis Involved?</th><th>What It Does</th></tr></thead><tbody><tr><td>**Tangent space normal maps**</td><td>TBN basis</td><td>Converts normals from UV space to world space</td></tr><tr><td>**Custom pivot control**</td><td>Custom basis</td><td>Reorients local X/Y/Z axes</td></tr><tr><td>**Surface alignment shader**</td><td>Basis from world normals</td><td>Builds a local space aligned to geometry</td></tr><tr><td>**Procedural animation**</td><td>Create and apply basis</td><td>Bones, aim constraints, etc.</td></tr><tr><td>**Billboarding**</td><td>Rebuild basis using camera</td><td>Rotate object to face view direction</td></tr></tbody></table>

---

##   


> ✅ A **basis** is a set of vectors that define a space's axes.  
> A **matrix = basis + position.**  
> You use it to **move**, **rotate**, or **remap** things between spaces.

- **Identity Matrix**
    
    
    - Does nothing when applied to a vector.
    - Basis for understanding other transformations.
    - Represents standard world coordinates.
- **Uniform Scaled Matrix**
    
    
    - Modify all diagonal values equally.
    - Scales objects equally in all directions.
    - E.g., `0.5` on all diagonal entries = mesh is scaled down by half.
- **Axis Scaled Matrix**
    
    
    - Change individual diagonal values independently.
    - Scales the object along specific axes.
    - E.g., a 3 in the X-axis = stretches object 3x in X only.
- **Mirror (Reflection) Matrix**
    
    
    - Multiply a diagonal entry by `-1` to mirror on that axis.
    - E.g., `-1` in X = flips mesh on the X-axis.
- **Matrix with Normalized Basis**
    
    
    - Basis vectors (matrix columns) are unit length and orthogonal.
    - Often used for pure rotation without scaling.
    - Each axis remains perpendicular and consistent in size.
- **Matrix with Non-Orthogonal Basis**
    
    
    - Columns are not perpendicular.
    - Causes skewing—object axes are no longer at right angles.
- **Matrix with Non-Normalized Basis**
    
    
    - Basis vectors may differ in length.
    - Can scale along arbitrary axes (not just X, Y, Z).
- **Rotation Matrices**
    
    
    - Represented by changing off-diagonal values (not just diagonal).
    - Rotate vectors in space while maintaining their magnitude.
- **Matrix Interpretation Tip**
    
    
    - Think of each matrix column as defining a new space axis.
    - Multiplying a vector by a matrix expresses it in that new space.

Let me know if you’d like a diagram or Unity version of any of these!