Show that the set of vectors defined as directed line segments does not form a group...

Question

Show that the set of vectors defined as directed line segments does not form a group (1) with respect to scalar product (2) with respect to vector product.

Answer 1

Hint in both cases: directly check the four defining properties of a group operation until you hit one that is violated.

Answer 2

Suppose that the vectors are in $V=\mathbb{R}^3$.

1) The dot product is an operation that take two vectors and gives a scalar as a result so $V$ cannot be a group with respect such operation because the result is not in $V$.

2) the cross product gives a vector as a result, but it's not associative (see Jacobi identity) as required by the axioms of a group.