Answer
See the explanation below.
Work Step by Step
Let $u=(5,12), \quad v=(3,4)$, $\langle u, v \rangle=u\cdot v$
Then, we have
$\langle u, v \rangle = u_1v_1+u_2v_2=15+48=63$, $\| u \| =\sqrt{\langle u, u\rangle}=\sqrt{u_1^2+u_2^2}=\sqrt{25+144}=13$, $\| v \| =\sqrt{\langle v, v\rangle}=\sqrt{v_1^2+v_2^2}=\sqrt{9+16}=5$, $\| u+v \| =\| (8,16) \| =\sqrt{64+256}=\sqrt{320}$,
Now, we get
(a) Cauchy-Schwarz Inequality: $|\langle u, v \rangle|=63\leq \|u\|\|v\|=(13)(5)=65.$
(b) The triangle inequality: $\|u+v\|=\sqrt{320}=17.89\leq\|u\|+\|v\|=13+5=18$
