Dot product with Einstein summation convention (index notation)

Question

My professor, when proving:
$$a \cdot b = a_ib_i$$
wrote:
$$(a_i \hat{e}_i)\cdot(b_j \hat{e}_j) = (a_ib_j)(\hat{e}_i\hat{e}_j)$$

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My doubt is:

Are we allowed to treat the dot product as ordinary multiplication once we have written everything in index notations? Does the same go for cross product?

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1. The confusion is mainly because we have moved $b_j$ from one bracket to another.
   In my mind, We have moved $b_j$ across the dot. This feels wrong.

2. By 'ordinary multiplication', I mean the kind of multiplication that we use in algebra to denote the product of two variables. We can move variables from one bracket to another there.

Answer 1

What you have in the right-hand side of

$(𝑎_𝑖𝑒̂_𝑖)⋅(𝑏_𝑗𝑒̂_𝑗)=(𝑎_𝑖𝑏_𝑗)(𝑒̂_𝑖𝑒̂_𝑗)$

is not ordinary multiplication, because $𝑒̂_𝑖$ and $𝑒̂_𝑗$ are vectors. There you have to use the dot product.

Switching to the common notation we have:

$a=\sum_i a_i 𝑒̂_𝑖$

$b=\sum_j b_j 𝑒̂_j$

and $$ a \cdot b= \sum_{i,j} a_i b_j (𝑒̂_𝑖 \cdot 𝑒̂_j) $$ There we are using that the dot product is bilinear.