Show the commutative property with dot product

  • Thread starter mr_coffee
  • Start date
  • #1
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1
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.
 

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
Gold Member
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18
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 [itex]\mathbb{R}[/itex].
 
  • #3
Astronuc
Staff Emeritus
Science Advisor
19,959
3,490
(a1 i + b1 j) [itex]\bullet[/itex] (a2 i + b2 j) =

a1 i [itex]\bullet[/itex] a2 i + a1 i [itex]\bullet[/itex] b2 j + b1 j [itex]\bullet[/itex] a2 i + b1 j [itex]\bullet[/itex] b2 j =

remember i [itex]\bullet[/itex] i = j [itex]\bullet[/itex] j = 1, and i [itex]\bullet[/itex] j = j [itex]\bullet[/itex] i = 0,

then regroup and show something similar to

a * b = b * a

and this can be extended to 3 dimensions
 
  • #4
117
0
well, the dot product is a definition.

as such

[tex]
\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
[/tex]

so, what is [tex]\vec{b}\bullet\vec{a}[/tex]?
 
  • #5
1,629
1
Cool so all I really need to show is this? or doesn't this prove it yet? Thanks for the replies everyone
[tex]
\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
[/tex]
 
  • #6
Doc Al
Mentor
45,248
1,598
You need to show that
[tex]\vec{b}\bullet\vec{a} = \vec{a}\bullet\vec{b}[/tex]

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

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