The Dot Product

Property 4 of Dot Product

(xa)∙b=x(a∙b)=a∙(xb) x is a scalar

Property 5 of Dot Product

0∙a=0 (First zero is the zero vector and the second is the numerical value zero)

Property 3 of Dot Product

a∙(b+c)=a∙b+a∙c

Property 1 of Dot Product

a∙a=|a|²

Determining orthogonal vectors

a∙b=0

Dot Product

a∙b=a₁b₁+a₂b₂+a₃b₃

Property 2 of Dot Product

a∙b=b∙a

Angle between Vectors

a∙b=|a||b|cosΘ cosΘ=(a∙b)/(|a||b|)

Scalar Projection of b onto a

comp_a b=(a∙b)/|a|

Vector Projection

proj_a b=a(a∙b)/|a|²