Find $-3\vec u+4\vec v$ given $u=4\vec i + \vec j$ and $v=5 \vec i - 2\vec j$

Question

Find the vector $-3\vec u+4\vec v$ given vectors $$\vec u=4 \vec \imath + \vec \jmath$$ and $$v=5\vec \imath - \vec \jmath.$$

Write answer in the form $a \vec \imath + b \vec \jmath$.

Answer 1

We have ${\bf u} = 4{\bf i} + {\bf j}$ and ${\bf v} = 5{\bf i} - 2{\bf j}$. Then $$\begin{align}-3{\bf u} + 4{\bf v} &= -3(4{\bf i} + {\bf j}) + 4(5{\bf i} - 2{\bf j}) \\ &= -12{\bf i} - 3{\bf j} + 20{\bf i}-8{\bf j} \\ &=8{\bf i} - 11{\bf j} \;\;\;\;\square\end{align}$$