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 .