# Vectors

• Jun 12th 2013, 10:49 PM
Fratricide
Vectors
ORST is a parallelogram. U is the midpoint of RS and V is the midpoint of ST. Relative to
the origin O, r, s, t, u and v are the position vectors of R, S, T, U and V respectively.
(a) Express s in terms of r and t.
(b) Express v in terms of s and t.
(c) Hence or otherwise show that 4 (u + v) = 3 (r + s + t)

I got (a) alright, but (b) and (c) are perplexing. I know that the answer for (b) is v = (1/2)(s + t), but I'm struggling to figure out how to get there.

Let's say \displaystyle \begin{align*} S = (S_x , S_y ) \end{align*} and \displaystyle \begin{align*} T = (T_x , T_y ) \end{align*}. Since V is the midpoint, that means
\displaystyle \begin{align*} V &= \left( \frac{ S_x + T_x }{ 2 } , \frac{ S_y + T_y }{ 2 } \right) \\ &= \frac{1}{2} \left( S_x + T_x , S_y + T_y \right) \\ &= \frac{1}{2} \left[ \left( S_x , S_y \right) + \left( T_x , T_y \right) \right] \\ &= \frac{1}{2} \left( S + T \right) \end{align*}