Question

Prove that the set of all vectors in a plane over the field of real numbers is a vector space

Answer 1

A vector space V over a field F is a nonempty set on which two operations are defined - addition and scalar multiplication.

for vector $u,v\isin P$ (plane):

$u+v=(u_1,u_2,u_3)+(v_1,v_2,v_3)=(u_1+v_1,u_2+v_2,u_3+v_3)\isin P$

$au=(au_1,au_2,au_3)\isin P$

So, the set of all vectors in a plane over the field of real numbers is a vector space.