Determine the vectors $\hat u+\hat v$ so that $$\hat u+\hat v=u$$ $$2\hat u+3\hat v=v$$ are true.
I tried: $$\hat{u} + \hat{v} = (1,0)$$ $$2\hat{u} + 3\hat{v} = (0,1)$$ $$(a,b) + (c,d) = (1,0)$$ $$2(a,b) + 3(c,d) = (0,1)$$ $$a + c = 1$$ $$b + d = 0$$ $$2a + 3c = 0$$ $$b + d = 1$$
This is obviously not true since $b+d$ cannot both equal 1 and 0 at the same time. Why does my solution not work and how do I solve it?
When we multiply a vector by a scalar number, like in $2(a,b)$, we have to multiply each coordinate, so we get $(2a,2b)$ and not $(2a,b)$.
Thus, your last equation should rather be $2b+3d=1$.