Show the commutative property with dot product

[mr_coffee]

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]

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]

(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

 

[teclo]

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 $$\vec{b}\bullet\vec{a}$$?

 

[mr_coffee]

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]

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.