Question

ABCD is a square and P, Q are the midpoints of BC, CD respectively. If AP = a and AQ = b, find in terms of a and b, the directed line segments (i) AB, (ii) AD, (iii) BD and (iv) AC. 

Answer 1

Let's place the coordinate center at the point A and direct the x and y axis along AB and AD. If the square side is $l$, then the coordinates of points are : $A=(0,0), B=(l,0), C=(l,l), D=(0,l), P=(l,\frac{l}{2}),Q=(\frac{l}{2},l)$. Therefore we have $a=(l,\frac{l}{2}), b=(\frac{l}{2}, l)$.

1. $\vec{AB }= \alpha a+\beta b$, $\begin{cases} \alpha l + \beta l/2 = l \\ \alpha l/2 + \beta l = 0 \end{cases}$, by solving we find $\vec{AB} = \frac{4}{3}a-\frac{2}{3}b$
2. $\vec{AD }= \alpha a+\beta b$, $\begin{cases} \alpha l + \beta l/2 = 0 \\ \alpha l/2 + \beta l = l \end{cases}$, by solving we find $\vec{AD} = -\frac{2}{3}a+\frac{4}{3}b$
3. $\vec{BD}=\vec{AD}-\vec{AB} = 2b-2a$
4. $\vec{AC} = \vec{AB}+\vec{BC} = \vec{AB}+\vec{AD} = \frac{2}{3}(a+b)$, as $BC = AD=(0,l)$.