Dot Product and Simplification

Question

How to simplify the expression

a b . x/((a b) . x)

Where a is a scalar and b and x are vectors.

I have tried the usual Simplify, FullSimplify, TensorExpand etc. but the expression does not simplify to 1.

Answer 1

You'll want to use TensorReduce or TensorExpand with appropriate assumptions for reducing expressions involving Dot:

TensorReduce[a b . x / ((a b) . x), 
 Assumptions -> {a \[Element] Reals, (b | x) \[Element] Vectors[3, Reals]}]

(* 1 *)

Answer 2

Use TensorExpand with reaonable Assumptions

    In[28]:= Assuming[(x | z) \[Element] Vectors[dim], 
 TensorExpand[{(a x)  \[TensorProduct] (q z ), (a x) . (q z + c  x)} ]]

Out[28]= {a q x\[TensorProduct]z, a c x . x + a q x . z}