Answer
$\textbf a+ \textbf b = -\textbf i+ \textbf j +2\textbf k$
$2\textbf a+ 3\textbf b = -4\textbf i+\textbf j +9\textbf k$
$|\textbf a| =\sqrt {14}$
$|\textbf {a-b}| =\sqrt {82}$
Work Step by Step
$\textbf a+ \textbf b = (\textbf i+ 2\textbf j -3\textbf k) + (-2\textbf i-\textbf j +5\textbf k) = -\textbf i+ \textbf j +2\textbf k$
$2\textbf a+ 3\textbf b = 2 (\textbf i+ 2\textbf j -3\textbf k) + 3(-2\textbf i-\textbf j +5\textbf k) = (2\textbf i+ 4\textbf j -6\textbf k) +(-6\textbf i-3\textbf j +15\textbf k) = -4\textbf i+\textbf j +9\textbf k$
$|\textbf a| = \sqrt {a_1^2 + a_2^2 + a_3^2} = \sqrt {1^2 + 2^2 + (-3)^2} = \sqrt {1 + 4 + 9} = \sqrt {14}$
$|\textbf {a-b}| = \sqrt {(a_1 - b_1)^2 + (a_2 - b_2)^2 + (a_3 - b_3)^2} = \sqrt {(1-(-2))^2 + (2 -(- 1))^2 + (-3-5)^2} = \sqrt {3^2 + 3^2 + (-8)^2} = \sqrt {9 + 9 + 64} = \sqrt {82}$
