"Find the coordinates of the point which divides the line segment joining A(2,3,5) to B(3, -1, 1) externally in the ratio 2 : 5. "

How does one solve this problem? And what does "externally" mean?

2. Hello karldiesen
Originally Posted by karldiesen
"Find the coordinates of the point which divides the line segment joining A(2,3,5) to B(3, -1, 1) externally in the ratio 2 : 5. "

How does one solve this problem? And what does "externally" mean?
Look at the attached diagram. $\displaystyle P_1$ divides $\displaystyle AB$ internally in the ratio $\displaystyle 2:5$ if $\displaystyle AP_1:P_1B = 2:5$; and $\displaystyle P_2$ divides $\displaystyle AB$ externally in the ratio $\displaystyle 2:5$ if $\displaystyle P_2A:P_2B = 2:5$; or (taking account of the directions) $\displaystyle AP_2:P_2B = -2:5$.

Now if points $\displaystyle A$ and $\displaystyle B$ have position vectors $\displaystyle \textbf{a}$ and $\displaystyle \textbf{b}$ respectively, then the point $\displaystyle P$ that divides the line segment $\displaystyle AB$ in the ratio $\displaystyle \lambda:\mu$ has position vector
$\displaystyle \textbf{p}=\frac{\mu\textbf{a}+\lambda\textbf{b}}{ \lambda + \mu}$
(It's quite straightforward to prove this: begin with $\displaystyle \textbf{AB} = \textbf{b}-\textbf{a}$, and then say $\displaystyle \textbf{p}= \textbf{a}+\frac{\lambda}{\lambda+\mu}\textbf{AB}$ ... etc)

So you can simply plug the position vectors of $\displaystyle A$ and $\displaystyle B$ into this formula with $\displaystyle \lambda = -2$ and $\displaystyle \mu = 5$. If my arithmetic is correct, I reckon the answer is $\displaystyle (\tfrac43, \tfrac{17}{3},\tfrac{23}{3})$.