Dot Product of Vectors
What is the Dot Product?
The dot product (also called the scalar product) is an operation that takes two vectors and returns a single number (a scalar). It tells us how much two vectors "align" with each other.
Unlike vector addition, which gives you another vector, the dot product gives you just a number — and that number carries deep geometric meaning!
📐 Formal Definition
Algebraic Definition
Given two vectors in 2D:
$\vec{a} = \langle a_1, a_2 \rangle$ and $\vec{b} = \langle b_1, b_2 \rangle$
The dot product is defined as:
You multiply corresponding components and then add the results.
Geometric Definition
The dot product can also be expressed using the angle between the two vectors:
where $|\vec{a}|$ and $|\vec{b}|$ are the magnitudes (lengths) of the vectors, and $\theta$ is the angle between them.
🔑 Key Properties
| Property | Formula | Meaning |
|---|---|---|
| Commutative | $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ | Order doesn't matter |
| Distributive | $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$ | Works over addition |
| Scalar Mult. | $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b})$ | Scalars factor out |
| Self Dot Product | $\vec{a} \cdot \vec{a} = |\vec{a}|^2$ | Gives magnitude squared |
🧭 What Does the Sign Tell Us?
Positive: $\vec{a} \cdot \vec{b} > 0$
Angle $\theta < 90°$
Vectors point in similar directions
Zero: $\vec{a} \cdot \vec{b} = 0$
Angle $\theta = 90°$
Vectors are perpendicular (orthogonal)