Answer
see the proof below.
Work Step by Step
For any $(u_1,u_2), (v_1, v_2 ), (w_1, w_2) \in {R}^2 , k \in {R}$
(1) $\langle (u_1,u_2), (u_1, u_2 )\rangle=u_1^2+5u_2^2>0$ and $\langle (u_1,u_2), (u_1, u_2 )\rangle=u_1^2+5u_2^2=0$ if and only if $u_1=0$, $u_2=0$.
(2) \begin{align*}
\langle (u_1,u_2), (v_1, v_2 )\rangle&=u_1v_1+5u_2v_2\\
&=v_1u_1+5v_2u_2\\
&=\langle (v_1, v_2 ),(u_1,u_2)\rangle.
\end{align*}
(3) \begin{align*}
\langle (ku_1,ku_2), (v_1, v_2 )\rangle&=3ku_1v_1+ku_2v_2\\
&=k(u_1v_1+5u_2v_2)\\
&=k\langle (u_1,u_2), (v_1, v_2 )\rangle.
\end{align*}
(4) \begin{align*}
\langle (u_1,u_2)+(v_1, v_2 ), (w_1, w_2)\rangle&=\langle (u_1+v_1, u_2+ v_2 ),(w_1, w_2)\rangle\\
&=(u_1+v_1)w_1+5(u_2+ v_2)w_2\\
&=u_1w_1+v_1w_1+5u_2w_2+ 5v_2w_2\\
&=(u_1w_1+5u_2w_2)+(v_1w_1+5 v_2w_2)\\
&=\langle (u_1, u_2 ),(w_1,w_2)\rangle+\langle (v_1, v_2 ),(w_1,w_2)\rangle.
\end{align*}
