Answer
$2\vec{u}=2\vec{i}+2\vec{j}$
$-3\vec{v}=-3\vec{i}+3\vec{j}$
$\vec{u}+\vec{v}=2\vec{i}$
$3\vec{u}-4\vec{v}=-\vec{i}+7\vec{j}$
Work Step by Step
$\vec{u}=\vec{i}+\vec{j},$ $\vec{v}=\vec{i}-\vec{j}$
$2\vec{u}$
Multiply both components of the $\vec{u}$ by $2$:
$2\vec{u}=2(\vec{i}+\vec{j})=2\vec{i}+2\vec{j}$
$-3\vec{v}$
Multiply both components of $\vec{v}$ by $-3$:
$-3\vec{v}=-3(\vec{i}-\vec{j})=-3\vec{i}+3\vec{j}$
$\vec{u}+\vec{v}$
Add the same component from both vectors together and simplify:
$\vec{u}+\vec{v}=(\vec{i}+\vec{j})+(\vec{i}-\vec{j})=(1+1)\vec{i}+(1-1)\vec{j}=2\vec{i}$
$3\vec{u}-4\vec{v}$
Multiply $\vec{u}$ by $3$ and $\vec{v}$ by $-4$, then add the two vectors together and simplify:
$3\vec{u}-4\vec{v}=3(\vec{i}+\vec{j})-4(\vec{i}-\vec{j})=(3\vec{i}+3\vec{j})+(-4\vec{i}+4\vec{j})=...$
$...=(3-4)\vec{i}+(3+4)\vec{j}=-\vec{i}+7\vec{j}$
