Dot product

Summary

• If $u=\left( u_1,⋯,u_n \right)$ and $v=\left( v_1,⋯,v_n \right)$ are two vectors then the dot product $u⋅v$ is defined as the number

$$u⋅v =u_1v_1+…+u_nv_n=\sum_{i=1}^{n}{ u_iv_i }$$

• If $u,v,w$ are $n$ -vectors and $a,β$ are scalars then:
• $u⋅v=v⋅u$
• $u⋅(v+w)=u⋅v+u⋅w$
• $(αu)⋅v=α(u⋅v)=u⋅(αv)$
• $(αu)⋅(βv)=αβ(u⋅v)$
• The dot product is also called the inner product or the scalar product .