Q14 (a) (iii)

Criteria	Marks
Provides correct solution	2
Writes $\overrightarrow{SL}$ or $\overrightarrow{BL}$ as a linear combination of $\overrightarrow{SB}$ and $\overrightarrow{SC}$, or equivalent merit	1

Sample answer:
$L$ is on the line $SK$ so $\overrightarrow{SL}=\lambda\overrightarrow{SK}$ for some real number $\lambda$.
$\overrightarrow{SL}=\frac\lambda4\overrightarrow{SB}+\frac\lambda3\overrightarrow{SC}$
Similarly, there exists a real number $\mu$ such that
$\overrightarrow{BL}=\mu\overrightarrow{BC}=-\mu\overrightarrow{SB}+\mu\overrightarrow{SC}$

$$\overrightarrow{SL}=\overrightarrow{SB}+\overrightarrow{BL}=(1-\mu)\overrightarrow{SB}+\mu\overrightarrow{SC}$$ $$\frac\lambda4\overrightarrow{SB}+\frac\lambda3\overrightarrow{SC}=(1-\mu)\overrightarrow{SB}+\mu\overrightarrow{SC}$$

$\overrightarrow{SB}$ and $\overrightarrow{SL}$ are not parallel, so from part (ii),
$\frac\lambda4=1-\mu\quad\text{and}\quad\frac\lambda3=\mu$
$\lambda=3\mu,\quad\lambda=4-4\mu$
$3\mu=4-4\mu,\quad\mu=\frac47$
$\therefore\overrightarrow{BL}=\mu\overrightarrow{BC}=\frac47\overrightarrow{BC}$