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Math Help - 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. 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}).

    Grandad
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