Answer
a) $\textbf a+ \textbf b = \left< 2, -18 \right>$
b) $2\textbf a+ 3\textbf b = \left< 1, -42 \right>$
c)$|\textbf a| = 13$
d)$|\textbf {a-b}| =10$
Work Step by Step
$\textbf a+ \textbf b = \left< 5, -12 \right> + \left< -3, -6 \right> = \left< 5 + (-3), -12 + (-6) \right> = \left< 2, -18 \right>$
$2\textbf a+ 3\textbf b = 2\left< 5, -12 \right> + 3\left< -3, -6 \right> = \left< 10, -24 \right> + \left< -9, -18 \right> = \left< 1, -42 \right>$
$|\textbf a| = \sqrt {a_1^2 + a_2^2} = \sqrt {5^2 + (-12)^2} = \sqrt {25 + 144} = \sqrt {169} = 13$
$|\textbf {a-b}| = \sqrt {(a_1 - b_1)^2 + (a_2 - b_2)^2} = \sqrt {(5-(-3))^2 + (-12 - (-6))^2} = \sqrt {8^2 + (-6)^2} = \sqrt {64 + 36} = \sqrt {100} = 10$
