# Show the commutative property with dot product

Hello everyone, does anyone know the proof of the dot products communative property (a)(b) = (b)(a) or any websites that show the dot products communative property? or other properties? Thanks! The book only shows the distributed property.

quasar987
Homework Helper
Gold Member
Write a.b and then b.a in terms of the components or as |a||b|cosO. Commutativity of the dot products follows from commutativity of addition (resp. multiplication) in $\mathbb{R}$.

Astronuc
Staff Emeritus
(a1 i + b1 j) $\bullet$ (a2 i + b2 j) =

a1 i $\bullet$ a2 i + a1 i $\bullet$ b2 j + b1 j $\bullet$ a2 i + b1 j $\bullet$ b2 j =

remember i $\bullet$ i = j $\bullet$ j = 1, and i $\bullet$ j = j $\bullet$ i = 0,

then regroup and show something similar to

a * b = b * a

and this can be extended to 3 dimensions

well, the dot product is a definition.

as such

$$\vec{a}\bullet\vec{b}= \left( \begin{array}{cc} a_x \\ a_y\\ a_z \end{array} \right) \bullet \left( \begin{array}{cc} b_x \\ b_y\\ b_z \end{array} \right) =a_xb_x + a_yb_x + a_zb_z$$

so, what is $$\vec{b}\bullet\vec{a}$$?

Cool so all I really need to show is this? or doesn't this prove it yet? Thanks for the replies everyone
$$\vec{b}\bullet\vec{a}= \left( \begin{array}{cc} b_x \\ b_y\\ b_z \end{array} \right) \left( \begin{array}{cc} a_x \\ a_y\\ a_z \end{array} \right) \bullet =b_xa_x + b_xa_y + b_za_z$$

Doc Al
Mentor
You need to show that
$$\vec{b}\bullet\vec{a} = \vec{a}\bullet\vec{b}$$

Evaluate each side and compare. Make use of the commutivity of ordinary addition, as quasar987 advised.