The problem asks to express the vectors $\overline{MK}$, $\overline{NL}$, $\overline{NK}$, and $\overline{KN}$ in terms of vectors $\overline{KL} = a$, $\overline{LM} = b$, and $\overline{NM} = 3a$, using vector addition.

1. Problem Description

The problem asks to express the vectors $\overline{MK}$, $\overline{NL}$, $\overline{NK}$, and $\overline{KN}$ in terms of vectors $\overline{KL} = a$, $\overline{LM} = b$, and $\overline{NM} = 3a$, using vector addition.

2. Solution Steps

a) Express $\overline{MK}$ in terms of $a$ and $b$.

$\overline{MK} = \overline{ML} + \overline{LK}$

$\overline{MK} = -b - a = -(a+b)$

b) Express $\overline{NL}$ in terms of $a$ and $b$.

$\overline{NL} = \overline{NM} + \overline{ML}$

$\overline{NL} = 3a - b$

c) Express $\overline{NK}$ in terms of $a$ and $b$.

$\overline{NK} = \overline{NM} + \overline{ML} + \overline{LK}$

$\overline{NK} = 3a - b - a = 2a - b$

d) Express $\overline{KN}$ in terms of $a$ and $b$.

$\overline{KN} = \overline{KL} + \overline{LM} + \overline{MN}$

$\overline{KN} = a + b - 3a = -2a + b = b - 2a$

3. Final Answer

a) $\overline{MK} = -(a+b)$

b) $\overline{NL} = 3a - b$

c) $\overline{NK} = 2a - b$

d) $\overline{KN} = b - 2a$