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Thread: Problem about vectors:

  1. #1
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    Problem about vectors:

    "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?
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  2. #2
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    Hello karldiesen
    Quote Originally Posted by karldiesen View Post
    "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 $\displaystyle (\tfrac43, \tfrac{17}{3},\tfrac{23}{3})$.

    Grandad
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