Answer
see the proof below.
Work Step by Step
For any $(u_1,u_2,u_3), (v_1, v_2,v_3 ), (w_1, w_2,w_3) \in {R}^3 , k \in {R}$
(1) $\langle (u_1,u_2,u_3), (u_1, u_2 ,u_3)\rangle= 2u_1^2+ u_2^2+2u_3^2>0$ and $\langle(u_1,u_2,u_3), (u_1, u_2 ,u_3)\rangle= 2u_1^2+u_2^2+2u_3^2=0$ if and only if $u_1=0$, $u_2=0$, $u_3=0$.
(2) \begin{align*}
\langle (u_1,u_2,u_3), (v_1, v_2,v_3 )\rangle&= 2u_1v_1+u_2v_2+2u_3v_3\\
&=2 v_1u_1+v_2u_2+2v_3u_3\\
&=\langle (v_1, v_2,v_3 ),(u_1,u_2,u_3)\rangle.
\end{align*}
(3) \begin{align*}
\langle (ku_1,ku_2,ku_3), (v_1, v_2 ,v_3)\rangle &=2ku_1v_1+ku_2v_2+2ku_3v_3\\
&=k( 2u_1v_1+u_2v_2+2u_3v_3)\\
&=k\langle(u_1,u_2,u_3), (v_1, v_2,v_3 )\rangle.
\end{align*}
(4)
\begin{aligned}
&\left\langle\left(u_{1}, u_{2}, u_{3}\right)+\left(v_{1}, v_{2}, v_{3}\right),\left(w_{1}, w_{2}, w_{3}\right)\right\rangle =\left\langle\left(u_{1}+v_{1}, u_{2}+v_{2}, u_{3}+v_{3}\right),\left(w_{1}, w_{2}, w_{3}\right)\right\rangle \\ &=2\left(u_{1}+v_{1}\right) w_{1}+\left(u_{2}+v_{2}\right) w_{2}+2\left(u_{3}+v_{3}\right) w_{3} \\ &=\left(2u_{1} w_{1}+ u_{2} w_{2}+2u_{3} w_{3}\right)+\left(2v_{1} w_{1}+ v_{2} w_{2}+2v_{3} w_{3}\right) \\ &=\left\langle\left(u_{1}, u_{2}, u_{3}\right),\left(w_{1}, w_{2}, w_{3}\right)\right\rangle+\left\langle\left(v_{1}, v_{2}, v_{3}\right),\left(w_{1}, w_{2}, w_{3}\right)\right\rangle \end{aligned}
