The discussion centers on finding the position vector of point P, which divides the line segment AB in the ratio 5:-2, with points A (-1, 6, 4) and B (4, 1, -1). The participants express confusion regarding the interpretation of negative ratios in vector division and clarify that the notation 'P divides AB in the ratio r:s' implies the relationship $$\frac{\vec{AP}}{r}=\frac{\vec{PB}}{s}$$. The negative ratio indicates that point P is positioned outside the segment AB, leading to further inquiries about the implications of such terminology in vector mathematics.
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andrewkirk said:What strange terminology! I've not seen anybody use ratios with vectors that way before.
I can only guess that when they say 'P divides AB in the ratio r:s' they mean
$$\frac{\vec{AP}}{r}=\frac{\vec{PB}}{s}$$
Which of your two options does that favour?