Calc III, Showing the dot product is communative, help plz

[mr_coffee]

Hello everyone. I'm trying to show ab = ba. //communative property of dot product
This is what I have, is it enough to show this?
ab = (ax, ay, az) (bx, by, bz) = axbx + ayby + azbz;
ab = (axi + ayj) (bxi + byj); //note: i and j are unit vectors
= axbx(i)(i) + ayby(j)(j) + axby(i)(j) + aybx(j)(i)
= axbx+ayby

ba = (bx, by, bz) (ax,ay,az) = bxax + byay + bzaz;

ba = (bxi + byj) (axi + ayj)
= bxax(i)(i) + byay(j)(j) + bxay(i)(j) + byax(j)(i)
= bxax + byay

Is this enough to show the dot product is communative?
Any help would be great if it isn't!

 

[pka]

Sorry to say but your notation is not standard!
a=<a<SUB>1</SUB>,a<SUB>2</SUB>,a<SUB>3</SUB>> & b=<b<SUB>1</SUB>,b<SUB>2</SUB>,b<SUB>3</SUB>>
a•b= a<SUB>1</SUB>b<SUB>1</SUB>+ a<SUB>2</SUB>b<SUB>2</SUB>+ a<SUB>3</SUB>b<SUB>3</SUB>.
Because the real numbers are commutative, the proof is trivial.