Suppose that O A and B are three noncollinear points in a pl

Suppose that O, A and B are three non-collinear points in a plane. Let OP = 2 OB - OA, OQ = 20 A - 30B, O B = 3 OB - OA and OS = OA + OB. Write a vector equation for the line that passes through Q and is parallel to the vector OP, in terms of the vectors OA and OB. Determine whether or not the point R lies on the line from part (a). Determine whether or not the point S lies on the line from part (a). Show algebraically that the vectors OS and OP -3/4 OR are linearly dependent. Find the point of intersection of the line through O and B and the line from part (a). Show that the line through O and P does not intersect the line from part (a).

Solution

(a) the eqation of the line passes through Q and parallel through OP is r=OQ+t(OP) r=2 OA-3 OB+t(OB-OA) =(2-t)OA+(-3+2t)OB------------(a) (b)if the point R lies on (a)then OR= (2-t)OA+(-3+2t)OB 3OB-OA=(2-t)OA+(-3+2t)OB 2-t=-1,t=3 and -3+2t=3 ,t=3 R LIES ON (a) (c) if S LIES ON (a) then OS=(2-t)OA+(-3+2t)OB OA+OB=(2-t)OA+(-3+2t)OB 2-t=1,-3+2t=1 t=1 and t=2 are not uniqe S DOES NOT LIES ON (a) (d)OP-3OR/4=(4OP-3OQ)/4 4OP-3OR=4(2OB-OA)-3(3OB-OA)       =(-OA-OB) ( -4OP+3OR)= OA+OB ( -4OP+3OR)/(-4+3)=OS OS DIVIDES THE LINE PR IN THE RATIO -4:3 THEY ARE COLLINEAR.HENCE OP-(3\\4)OR,OS ARE DEPENDENT. (E) an the eqation of the line passes through Od B IS r=(1-S)O+sOB IF IT INTERSECTS(a) then (1-s)O+sOB=(2-t)OA+(-3+2t)OB 1-s=0,-3+2t=s s=1,t=2 the point of intersection is OB (f) an the eqation of the line passes throughO AND P IS r=(1-s)O+s OP=s(2OB-OA) if it intersects (a) then 2sOB-sOA=(2-t)OA+(-3+2t)OB- 2-t=-s,2s=-3+2t t-s=2,2t-2s=-3 have no solutions. hence OP DOES NOT INTERCST THE LINE(a)