Vector Products: Dot, Cross, and Geometry

The Dot Product: A Measure of Alignment

The first type of vector multiplication we'll explore is the dot product. Unlike the operations you've learned so far, the dot product takes two vectors and returns a single number—a scalar. This scalar value tells us something incredibly useful about the relationship between the two vectors.

Calculating the Dot Product

The calculation is straightforward: you multiply the corresponding components of the two vectors and then sum the results.

For two 3D vectors $\mathbf{a}=(a_x,a_y,a_z)$ and $\mathbf{b}=(b_x,b_y,b_z)$, the dot product is:

Let's start with a formal definition and a few examples.

Geometric Interpretation of the Dot Product

The calculation is simple, but the real power of the dot product comes from its geometric meaning. It essentially measures how much one vector "points along" the other.

The sign of the dot product gives us a quick way to classify the angle between two vectors:

• If $\mathbf{a}\cdot \mathbf{b}>0$, the vectors are pointing in a similar direction (the angle between them is less than 90°).
• If $\mathbf{a}\cdot \mathbf{b}=0$, the vectors are perpendicular (the angle is exactly 90°). This is a very important property.
• If $\mathbf{a}\cdot \mathbf{b}<0$, the vectors are pointing in generally opposite directions (the angle is greater than 90°).

This interpretation is directly related to another definition of the dot product:

Here, $\Vert \mathbf{a}\Vert$ and $\Vert \mathbf{b}\Vert$ are the magnitudes (lengths) of the vectors, and $\theta$ (theta) is the angle between them. This relationship is the key to finding the angle between any two vectors.

The following reading explains these geometric concepts in detail.

To see a practical worked example of finding the angle between two vectors, watch the following video clip.

Application: The Dot Product in C

In C, we can write a simple function to compute the dot product, which we'll add to our growing math library.

// Calculates the dot product of two vectors
float vec3_dot(Vec3 a, Vec3 b) {
    return a.x * b.x + a.y * b.y + a.z * b.z;
}

We'll use this function frequently for tasks like:

• Basic AI: To check if an enemy is in a character's cone of vision.
• Lighting: To determine how much light hits a surface by comparing the surface normal to the light direction. This is a fundamental concept we'll explore in detail in Module 6.

The Cross Product: Creating a Perpendicular Vector

Where the dot product gives us a scalar, the cross product is an operation between two 3D vectors that produces a new 3D vector. The key property of this new vector is that it is perpendicular (orthogonal) to both of the original vectors.

This operation is only defined for 3D vectors.

Geometric Interpretation of the Cross Product

Understanding the geometry of the cross product is essential. The 3Blue1Brown channel has a video that provides excellent intuition.

As the video explained, the cross product $\mathbf{a}\times \mathbf{b}$ gives a new vector with two defining properties:

1. Direction: It is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$. We determine which of the two perpendicular directions it points in using the right-hand rule. If you point your index finger along $\mathbf{a}$ and your middle finger along $\mathbf{b}$, your thumb will point in the direction of $\mathbf{a}\times \mathbf{b}$.
2. Magnitude: Its length, $\Vert \mathbf{a}\times \mathbf{b}\Vert$, is equal to the area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$. This is also given by the formula $\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \sin (\theta )$. This means if the vectors are parallel ($\theta =0^{\circ }$ or $180^{\circ }$), the cross product is the zero vector [0, 0, 0].

Calculating the Cross Product

The formula for the cross product is a bit more involved than for the dot product. For two vectors $\mathbf{a}=(a_x,a_y,a_z)$ and $\mathbf{b}=(b_x,b_y,b_z)$:

Note that the cross product is not commutative. In fact, it is anticommutative: $\mathbf{a}\times \mathbf{b}=-(\mathbf{b}\times \mathbf{a})$. The resulting vector points in the exact opposite direction.

The following reading provides the formula and discusses its primary application.

Application: Calculating Surface Normals

The most important use of the cross product in 3D graphics is to calculate a normal vector. A normal vector is a vector that is perpendicular to a surface. For a flat triangle defined by three points P1, P2, and P3, we can create two vectors along its edges:

• $\mathbf{u}=P2-P1$
• $\mathbf{v}=P3-P1$

The cross product $\mathbf{n}=\mathbf{u}\times \mathbf{v}$ will give us the normal vector for that triangle. Normal vectors are crucial for calculating how light should reflect off a surface.

Application: The Cross Product in C

Here is the C function for the cross product. It's a direct translation of the formula.

// Calculates the cross product of two vectors
Vec3 vec3_cross(Vec3 a, Vec3 b) {
    Vec3 result = {
        .x = a.y * b.z - a.z * b.y,
        .y = a.z * b.x - a.x * b.z,
        .z = a.x * b.y - a.y * b.x
    };
    return result;
}

Conclusion

In this lesson, we've added two incredibly useful tools to our 3D math toolkit. While their calculations seem abstract, their geometric interpretations are what make them indispensable for game development.

Key Takeaways:

• The dot product takes two vectors and returns a scalar. It measures their alignment.

  • result > 0: similar direction.
  • result = 0: perpendicular.
  • result < 0: opposite directions.
  • Primary Use: Calculating the angle between vectors, and for relational checks (e.g., lighting, AI).

• The cross product takes two 3D vectors and returns a new 3D vector.

  • The resulting vector is perpendicular to both input vectors.
  • Its magnitude is the area of the parallelogram formed by the input vectors.
  • Primary Use: Calculating surface normals and creating orthogonal coordinate axes.

In our next lesson, we will focus on vector magnitude (length) and the process of normalization—creating a vector with a length of 1. This is a crucial step that builds on today's concepts, allowing us to work with pure directions, which is fundamental to almost every aspect of 3D graphics.