This is the complete problem:
The solution:
I tried to find out why -v1' = (2/3)v1 + (1/3)v2 but I still don't get it. If anyone can help me, I'll be very grateful.
2026-03-28 00:06:19.1774656379
Change of basis: find the coordinates of the point relative to the coordinate system.
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To calculate $v_1'$ you have to find other vectors that compose $v_1'$. Since $\frac{|\overrightarrow{BD}|}{|\overrightarrow{DC}|} = \frac{1}{2}$ you get $v_1' = \frac{1}{3} \overrightarrow{CB} - v_1$. With $\overrightarrow{CB} = -v_2 + v_1$ you get $$v_1' = \frac{1}{3}(-v_2 + v_1) - v_1 = -\frac{1}{3}v_2 - \frac{2}{3} v_1$$