Answer
(a) $\langle -4,-2 \rangle$
(b) $\langle -13,4 \rangle$
(c) $\langle 3,5 \rangle$
Work Step by Step
(a) To find the value of $2\textbf{u}$, we substitute the vector $\textbf{u}$ in the expression and simplify:
$2\textbf{u}$
$=2\cdot\langle -2,-1 \rangle$
$=\langle 2(-2),2(-1) \rangle$
$=\langle -4,-2 \rangle$
(b) To find the value of $2\textbf{u}+3\textbf{v}$, we substitute the vectors $\textbf{u}$ and $\textbf{v}$ in the expression and simplify:
$2\textbf{u}+3\textbf{v}$
$=2\cdot\langle -2,-1 \rangle+3\cdot\langle -3,2 \rangle$
$=\langle 2(-2),2(-1) \rangle+\langle 3(-3),3(2) \rangle$
$=\langle -4,-2 \rangle+\langle -9,6 \rangle$
$=\langle -4-9,-2+6 \rangle$
$=\langle -13,4 \rangle$
(c) To find the value of $\textbf{v}-3\textbf{u}$, we substitute the vectors $\textbf{u}$ and $\textbf{v}$ in the expression and simplify:
$\textbf{v}-3\textbf{u}$
$=\langle -3,2 \rangle-3\cdot\langle -2,-1 \rangle$
$=\langle -3,2 \rangle-\langle 3(-2),3(-1) \rangle$
$=\langle -3,2 \rangle-\langle -6,-3 \rangle$
$=\langle -3-(-6),2-(-3) \rangle$
$=\langle 3,5 \rangle$
