Answer
$\textbf a+ \textbf b = 2\textbf i-2\textbf j +3\textbf k$
$2\textbf a+ 3\textbf b = 4\textbf i-2\textbf j +5\textbf k$
$|\textbf a| = 6$
$|\textbf {a-b}| = \sqrt {65}$
Work Step by Step
$\textbf a+ \textbf b = (2\textbf i-4\textbf j +4\textbf k) + (2\textbf j -\textbf k) = 2\textbf i-2\textbf j +3\textbf k$
$2\textbf a+ 3\textbf b = 2 (2\textbf i-4\textbf j +4\textbf k) + 3(2\textbf j -\textbf k) = (4\textbf i-8\textbf j +8\textbf k) + (6\textbf j -3\textbf k) = 4\textbf i-2\textbf j +5\textbf k$
$|\textbf a| = \sqrt {a_1^2 + a_2^2 + a_3^2} = \sqrt {2^2 + (-4)^2 + 4^2} = \sqrt {4 + 16 + 16} = \sqrt {36} = 6$
$|\textbf {a-b}| = \sqrt {(a_1 - b_1)^2 + (a_2 - b_2)^2 + (a_3 - b_3)^2} = \sqrt {(2-0)^2 + (-4 -2)^2 + (4-(-1))^2} = \sqrt {2^2 + (-6)^2 + 5^2} = \sqrt {4 + 36 + 25} = \sqrt {65}$
