dot product

Let $u=(u_1,u_2,\ldots,u_n)$ and $v=(v_1,v_2,\ldots,v_n)$ two vectors on $k^n$ where $k$ is a field (like $\mathbb{R}$ or $\mathbb{C}$ ). Then we define the dot product of the two vectors as: $$u\cdot v=u_1v_1+u_2v_2+\cdots+u_nv_n.$$

Notice that $u\cdot v$ is NOT a vector but a scalar (an element from the field $k$ ).

If $u,v$ are vectors in $\mathbb{R}^n$ and $\vartheta$ is the angle between them, then we also have $$u\cdot v=\Vert u\Vert\Vert v\Vert \cos\vartheta.$$ Thus, in this case, $u\perp v$ if and only if $u\cdot v=0$ .

The special case $u \cdot u$ of scalar product is the scalar square of the vector $u$ . In $\mathbb{R}^n$ it equals to the square of the length of $u$ : $$u \cdot u = \Vert u \Vert^2$$