$$\langle 13,31,-9 \rangle.$$
Work Step by Step
$$5\langle 2,2,-3 \rangle+3\langle 1,7,2 \rangle=\langle 10,10,-15 \rangle+\langle 3,21,6 \rangle\\ =\langle 13,31,-9 \rangle.$$
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