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

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

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
<br /> \vec{b}\bullet\vec{a}=<br /> <br /> \left(<br /> \begin{array}{cc}<br /> b_x \\<br /> b_y\\<br /> b_z<br /> \end{array}<br /> \right)<br /> \left(<br /> \begin{array}{cc}<br /> a_x \\<br /> a_y\\<br /> a_z<br /> \end{array}<br /> \right)<br /> \bullet<br /> <br /> =b_xa_x + b_xa_y + b_za_z<br />

 

[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.