$$a \cdot (a \cdot b)=(a \cdot a)(a \cdot b)$$
Is this identity true when $a$ and $b$ are vectors, and when $\cdot$ is the dot product operator? And assuming that $()()$ means multiplying the contents of the parentheses.
Can anyone please post explanation and links of this identity? I saw it used somewhere, but I cannot seem to find documentation of it in textbooks or in search engine research using the key words I could think of.
It is used in a suggested solution for the following problem:
"Show that the vector orth_a b = b - proj_a b is orthogonal to a. (It is called an orthogonal projection of b.)"
Where a and b are vectors, and proj_a b is the vector projection of b onto a.
Really what I need is to learn to answer this problem, and the part of the suggested proof I don't understand is given in my initial post above. If you have a better answer to this problem, I would love to learn how your approach works.