I don't know what the question is even asking. There is no notion of the "multiplicative inverse" of a vector (well... not in the sense of the vector space structure anyway).
Show why scalar multiplicative inverse property (∀a ≠ 0) (∃ b) ab=1 has no analogue using vector-by-scalar multiplication.
I'm not sure how to go about this. Help would be appreciated I know what the two symbols ∀ ∃ represent, but not sure what to do for this question.
1. If you have real numbers you know that for every number exists a number , called the reciprocal of a, such that .
2. If you have the scalar product of two vectors the result is not a vector but a single real number. This relation is not invertible.
As a practical consequence you are not allowed to calculate
But this question was about "vector by scalar multiplication" (which I take to mean scalar multiplication), not the dot product.
In order to have a multiplicative inverse, you must first have a multiplicative identity- but there is no vector, v, such that, for every scalar, a, av= a because the product is a vector, not a scalar. There is a scalar multiplicative identity, 1, so that 1v= 1 for every vector, v, but we cannot get that as the result of a multiplication because, again, the product is a vector, not a scalar.