Unit content
Dot product
The dot product combines two vectors into a scalar. For
$$\mathbf u=(u_1,\ldots,u_n),\qquad \mathbf v=(v_1,\ldots,v_n),$$
it is defined by
$$\mathbf u\cdot\mathbf v=u_1v_1+\cdots+u_nv_n.$$
For example,
$$(2,1)\cdot(3,-4)=6-4=2.$$
Length from the dot product
A vector dotted with itself gives the square of its length:
$$\mathbf v\cdot\mathbf v=|\mathbf v|^2.$$
Therefore
$$|\mathbf v|=\sqrt{\mathbf v\cdot\mathbf v}.$$
In $\mathbb R^2$, this reproduces the Pythagorean formula
$$|(x,y)|=\sqrt{x^2+y^2}.$$
Angle between vectors
For nonzero vectors, the dot product also satisfies
$$\mathbf u\cdot\mathbf v =|\mathbf u|,|\mathbf v|\cos\theta,$$
where $\theta$ is the angle between them. Hence
$$\cos\theta=\frac{\mathbf u\cdot\mathbf v}{|\mathbf u|,|\mathbf v|}.$$
A positive dot product corresponds to an acute angle, a negative one to an obtuse angle, and a zero dot product to a right angle.
The dot product therefore connects component arithmetic with length, angle and geometric alignment.