Hello kandyfloss Originally Posted by

**kandyfloss** The point C has position vector (2 3) and point D has position vector (1 2).Find the position vector of the point which divides CD in the ratio 4:-3

I recognized that the minus sign in 4:-3 indicates that it divides the line externally.However,i'm not able to get the ratios right when solving the problem.Can anyone show me with a diagram,the way the ratio is distributed in the line? (< i guess this is where i get it wrong?the ratio part)

Thanks

You're right in saying that the point divides the line segment $\displaystyle CD$ externally. If $\displaystyle CP:PD = 4 :-3$, then the ratio of the *distances* $\displaystyle CP$ and $\displaystyle PD$ is $\displaystyle 4:3$, but they are in *opposite directions *(hence the minus sign). So, from $\displaystyle C$ we shall go *out *$\displaystyle 4$ 'lengths' to get to $\displaystyle P$ and then *come back* $\displaystyle 3$ 'lengths' to $\displaystyle D$.

Take a look at the diagram I've attached. You'll see that:$\displaystyle CP:PD = 4:-3$

Now you probably know that the position vector of the point dividing the line joining the points with position vectors $\displaystyle \vec a$ and $\displaystyle \vec b$ in the ratio $\displaystyle \lambda:\mu$ is:$\displaystyle \frac{\mu\vec a + \lambda \vec b}{\lambda + \mu}$

So the point that divides $\displaystyle CD$ in the ratio $\displaystyle 4:-3$ has position vector:$\displaystyle \frac{-3\vec c + 4\vec d}{4-3}$$\displaystyle =-3\binom23+4\binom12$

$\displaystyle =\binom{-6}{-9}+\binom48$

$\displaystyle =\binom{-2}{-1}$

I hope that clears things up.

Grandad