I came across an exercise from the textbook The Fundamentals of Physics by Halliday which is as follows:
Which of the following are correct (meaningful) vector expressions? What is wrong with any incorrect expression?
a) $\vec{A} \cdot (\vec{B} \cdot \vec{C})$
The solution states that this is false, because "cannot dot a vector with a scalar".
However, I learned that I can multiply a vector with a scalar (by multiplying its magnitude with the scalar and rotating the vector by $\pi$ radians if the scalar is negative). As $\vec{B} \cdot \vec{C}$ yields a scalar, why should I not be able to multiply that scalar with the vector $\vec{A}$?
Or is there maybe a different notation for multiplying a vector with a scalar that I am missing?