dot product

$\mathbf u \cdot \mathbf v = u_1v_1 + \cdots + u_nv_n$ — a single number, not a vector.

For $\mathbf u = (u_1,\dots,u_n)$ and $\mathbf v = (v_1,\dots,v_n)$ in $\mathbb R^n$ , the dot product is defined entry by entry: $\mathbf u \cdot \mathbf v = u_1v_1 + u_2v_2 + \cdots + u_nv_n$ . Multiply matching entries, then add — the result is always a single real number (a scalar), never a vector. The dot product is commutative ( $\mathbf u\cdot\mathbf v = \mathbf v\cdot\mathbf u$ ), distributes over addition, and a vector dotted with itself, $\mathbf v\cdot\mathbf v = v_1^2+\cdots+v_n^2$ , is always $\ge 0$ , equal to $0$ only for $\mathbf v = \mathbf 0$ .