1. ## Vectors

Consider the following relationships between vectors $\displaystyle u$, $\displaystyle v$ and $\displaystyle w$.

i) $\displaystyle u = 2v + w$
ii) $\displaystyle w = v - u$

Which of the following statements is true?

A) $\displaystyle u = w$
B) $\displaystyle u = v$

C) $\displaystyle u = \frac{2}{3}v$

D) $\displaystyle u = \frac{3}{2}v$

E) $\displaystyle u = 3v$

Please explain why to make me understand.

2. Originally Posted by Mr Rayon
Consider the following relationships between vectors $\displaystyle u$, $\displaystyle v$ and $\displaystyle w$.

i) $\displaystyle u = 2v + w$
ii) $\displaystyle w = v - u$

Which of the following statements is true?

A) $\displaystyle u = w$

B) $\displaystyle u = v$

C) $\displaystyle u = \frac{2}{3}v$

D) $\displaystyle u = \frac{3}{2}v$

E) $\displaystyle u = 3v$

Please explain why to make me understand.
Note that substituting (ii) into (i) gives us $\displaystyle u=2v+v-u\implies 2u=3v\implies u=\tfrac{3}{2}v$ => (D) is true. Thus, its clear that (B), (C), and (E) are false.

Note that subsituting (i) into (ii) gives us $\displaystyle w=v-\left(2v+w\right)\implies w=-v-w\implies 2w=-v\implies 2w=-\tfrac{2}{3}u\implies 3w=-u$, which implies (A) is false.

Thus, (D) is the only one that is true.