Question

Check whether the vector $4\overline{i}+13\overline{j}-18\overline{k}$ is a linear combination of the vectors $\overline{i}-2\overline{j}+3\overline{k}$ and $2\overline{i}+3\overline{j}-4\overline{k}$.

Answer 1

Let if possible
$4\overline{i}+13\overline{j}-18\overline{k}$ = $x(\overline{i}-2\overline{i}+3\overline{k})+y(2\overline{i}+3\overline{j}-4\overline{k})$
$=(x+2y)\overline{i}+(-2x+3y)\overline{j}+(3x-4y)\overline{k}$
This is possible if the following equations are consistent.
$x+2y=4$ .....(1)
$-2x+3y=13$...(2)
$3x+4y=-18$....(3)
Solving equations (1) and (2) , we get
$\therefore x=-2,y=3$
These values satisfy equation (3) also
$\therefore 4\overline{i}+13\overline{j}-18\overline{k}=-2(\overline{i}-2\overline{j}+3\overline{k}+3(2\overline{i}+3\overline{j}-4\overline{k}\left)\right)$
$\therefore$ The vectors $4\overline{i}+13\overline{j}-18\overline{k}$ is a linear combination of the vectors $\overline{i}-2\overline{j}+3\overline{k}$ and $2\overline{i}=3\overline{j}-4\overline{k}$.