Answer
(a) $\langle -2,4 \rangle$
(b) $\langle 7,4 \rangle$
(c) $\langle 6,-6 \rangle$
Work Step by Step
(a) To find the value of $2\textbf{u}$, we substitute the vector $\textbf{u}$ in the expression and simplify:
$2\textbf{u}$
$=2\cdot\langle -1,2 \rangle$
$=\langle 2(-1),2(2) \rangle$
$=\langle -2,4 \rangle$
(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:
$2\textbf{u}+3\textbf{v}$
$=2\cdot\langle -1,2 \rangle+3\cdot\langle 3,0 \rangle$
$=\langle 2(-1),2(2) \rangle+\langle 3(3),3(0) \rangle$
$=\langle -2,4 \rangle+\langle 9,0 \rangle$
$=\langle -2+9,4+0 \rangle$
$=\langle 7,4 \rangle$
(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:
$\textbf{v}-3\textbf{u}$
$=\langle 3,0 \rangle-3\cdot\langle -1,2 \rangle$
$=\langle 3,0 \rangle-\langle 3(-1),3(2) \rangle$
$=\langle 3,0 \rangle-\langle -3,6 \rangle$
$=\langle 3-(-3),0-6 \rangle$
$=\langle 6,-6 \rangle$