Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:

\[(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right) \]

\[c \odot (x,y) = \left(c x,\,c y\right) .\]

(a) Show that vector addition is associative, that is:

\[\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right). \]

(b) Explain why \(V\) nonetheless is not a vector space.