So far, combining vectors gave back another vector. The dot product is different: it takes two vectors and returns a single number (a scalar). Multiply matching components and add:

For example \begin{bmatrix} 3 \\ 2 \end{bmatrix} \cdot \begin{bmatrix} 4 \\ 1 \end{bmatrix} = 3\cdot4 + 2\cdot1 = 14. That's the entire definition. It looks almost too plain to matter — yet this little number secretly measures how much two vectors point the same way, and it powers everything from neural networks to the geometry of angles.

Multiply and add

Set the two vectors below. The panel shows each component product and their sum — the dot product. Notice the sign: when the arrows roughly agree it's positive, when they roughly oppose it's negative, and somewhere in between it passes through zero.

The rules it obeys

The dot product is commutative (\vec{u}\cdot\vec{v} = \vec{v}\cdot\vec{u}) and distributes over addition (\vec{u}\cdot(\vec{v}+\vec{w}) = \vec{u}\cdot\vec{v} + \vec{u}\cdot\vec{w}). One special case is worth memorizing: a vector dotted with itself gives its length squared,

So the dot product already knows about length — and in the next lesson we'll see it knows about angle too.