"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. $P_1$ divides $AB$ internally in the ratio $2:5$ if $AP_1:P_1B = 2:5$; and $P_2$ divides $AB$ externally in the ratio $2:5$ if $P_2A:P_2B = 2:5$; or (taking account of the directions) $AP_2:P_2B = -2:5$.

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

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