School of Mathematics and Statistics, The University of Sydney
 5. Division of a line segment
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External division of a line segment

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We look at a situation similar to the one on the previous page, but allow m and n to have opposite sign. Again, R is the point such that -P--->R  1 = m- n-R-P-->    2, and we want to find the position vector of R (relative to O) in terms of the position vectors of P1 and P2.

P1                 P2                            R  O

Since ----> P1R is a negative multiple of ----> RP2, the vectors ----> P1R and ----> RP2 have opposite direction.

The point R then lies outside the line segment P1P2 (but still on the line joining P1 and P2). In these cases, R is said to divide the line segment P1P2 externally in the ratio m : n. The formula is, as in case of an internal division,

        ---->      ----> --->     nOP1  + m OP2 OR  =  ---------------,    m +  n /= 0.            m  + n

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