I was studying the dot product, and it says that (a.b).c makes no sense. so if you do (a.b) can = to β and then is it not possible to do β.c? WHY cant you 'dot' a scalar and a vector? why?
Because the dot product is defined ONLY for two vectors. You can multiply a vector by a scalar, and this is called scalar multiplication.
Have you tried it? Write down the definition of a dot product. Make up and write down three vectors and perform the calculation. (Note that definitions are made up, not proved. Can you prove that a cat is not a soda can? No. Its just not defined that way. Theorems and identities are what get proved, under the right definitions.)
The dot product, in two dimensions (for simplicity) is defined as: $$\vec{a}\cdot \vec{b}=a_xb_x+a_yb_y$$ Now, this assumes ##\vec{a}=(a_x,a_y)## and ##\vec{b}=(b_x,b_y)## are vectors. What would it mean to turn ##a## into a number? Certainly you can "define" the "dot product" of a scalar and a vector as: $$a\cdot\vec{b}=a\vec{b}=(ab_x,ab_y)$$ But that's just the same as a scalar product, so it would be supremely confusing to also call it a "dot product". That's why we don't call that the "dot product".