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The-points-A-B-and-C-have-position-vectors-a-b-and-c-respectively-reffrred-to-an-origin-O-i-Given-that-the-point-X-lie-on-AB-produced-so-that-AB-BX-2-1-find-x-the-position-vector-of-X-in-terms




Question Number 174489 by pete last updated on 02/Aug/22
The points A, B and C have position vectors  a, b and c respectively reffrred to an origin O.  i. Given that the point X lie on AB produced  so that AB : BX=2:1, find x, the position  vector of X in terms of b and c.  ii. if Y lies on BC, between B and C so that  BY : YC = 1:3, find y, the position vector  of Y in terms of b and c.  iii. Given that Z  is the mid point of AC,   show that X, Y and Z are collinear.
$$\mathrm{The}\:\mathrm{points}\:\mathrm{A},\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{have}\:\mathrm{position}\:\mathrm{vectors} \\ $$$$\boldsymbol{\mathrm{a}},\:\boldsymbol{\mathrm{b}}\:\mathrm{and}\:\boldsymbol{\mathrm{c}}\:\mathrm{respectively}\:\mathrm{reffrred}\:\mathrm{to}\:\mathrm{an}\:\mathrm{origin}\:\mathrm{O}. \\ $$$$\mathrm{i}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{point}\:\mathrm{X}\:\mathrm{lie}\:\mathrm{on}\:\mathrm{AB}\:\mathrm{produced} \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{AB}\::\:\mathrm{BX}=\mathrm{2}:\mathrm{1},\:\mathrm{find}\:{x},\:\mathrm{the}\:\mathrm{position} \\ $$$$\mathrm{vector}\:\mathrm{of}\:\mathrm{X}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\boldsymbol{\mathrm{b}}\:\mathrm{and}\:\boldsymbol{\mathrm{c}}. \\ $$$$\mathrm{ii}.\:\mathrm{if}\:\mathrm{Y}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{BC},\:\mathrm{between}\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{so}\:\mathrm{that} \\ $$$$\mathrm{BY}\::\:\mathrm{YC}\:=\:\mathrm{1}:\mathrm{3},\:\mathrm{find}\:{y},\:\mathrm{the}\:\mathrm{position}\:\mathrm{vector} \\ $$$$\mathrm{of}\:\mathrm{Y}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\boldsymbol{\mathrm{b}}\:\mathrm{and}\:\boldsymbol{\mathrm{c}}. \\ $$$$\mathrm{iii}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{Z}\:\:\mathrm{is}\:\mathrm{the}\:\mathrm{mid}\:\mathrm{point}\:\mathrm{of}\:\mathrm{AC},\: \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{X},\:\mathrm{Y}\:\mathrm{and}\:\mathrm{Z}\:\mathrm{are}\:\mathrm{collinear}. \\ $$

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