Showing dot product is commutative, need help please

[voidflux]

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!

[twanvl]

(Note: You should write dot product with a dot, not as normal multiplication.)

If you only work in three dimensions, something like this should be enough:

a.b
= (ax, ay, az).(bx, by, bz)       { defenition of a,b as 3d vector }
= ax bx + ay by + az bz           { defenition of dot product }
= ...                             { multiplication is commutative }
= ...
= b.a

I don't understand where you are going with the unit vectors:

ab = (ax, ay, az) (bx, by, bz) = axbx + ayby + azbz;
ab = (axi + ayj) (bxi + byj);

Where did z go?

[krtxmrtz]

Even much less is enough:

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

That's simply it! You're being asked to show the dot product is commutative, and this is obvious because the product of 2 numbers is commutative (you're not being asked to prove this!): axbx = bxax (etc), ax and bx being numbers (not vectors).