<s>[INST] Simplify the Dot Product in terms of $a$ and $b$ Where $a$ and $b$ are arbitrary vectors
$(a+2b) \cdot (2a-b)$
$$a\cdot(2a-b)+2b\cdot(2a-b) = 2(a\cdot a)-a\cdot b+1(b\cdot a)-2(b\cdot b)$$
$$=2(a)-ab+4ab-2(b)^2$$
$$=2a^2-2b^2$$
$$=2(a^2-b^2)$$
Where did i go wrong in simplifying this?
 [/INST] If you denote dot product as $\cdot$, you get
$$
(a+2b) \cdot (2a - b)
 = 2 a \cdot a + 4 b \cdot a - a \cdot b - 2 b \cdot b
 = 2 a \cdot a + 3 b \cdot a - 2 b \cdot b
$$
and you can further simplify $a\cdot a = |a|$ if you like.
Your error is the 2nd step (3rd line), where you assumed $$2b\cdot(2a-b) = 1 b\cdot a - 2 b \cdot b,$$ the the correct coefficient of $b \cdot a$ is 4, not 1.
</s>