On the xy plane where O is the origin, given points, `A(1, 0), B(0, 1) and C(1, 1)`. Let `P, Q, and R` be moving points on the line `OA, OB, OC` respectively such that `overline(OP)=45t overline((OA)),overline(OQ)=60t overline((OB)),overline(OR)=(1-t) overline((OC))` with `t>0.` If the three points `P,Q and R` are collinear then the value of `t` is equal to

On the xy plane where O is the origin, given points, A(1,0), B(0,1) and C(1,1). Let P,Q, and R be moving points on the line OA, OB, OC respectively such that vec(OP)=45t(vec(OA)), vec(OQ)=60t(vec(OB)), vec(OR)=(1-t)(vec(OC)) with t gt 0 . If the three points P,Q and R are collinear then the value of t is equal to

If the points P (0,3,2), Q(1,2,-2) and R (4,-1,t) are collinear, find the value of t.

Which of the following represents the volume of paralleloP1ped. If overline(OA)=overline(a), overline(OB)=overline(b), overline(OC)=overline(c) are its co-terminous edges?

ABDC is a parallelogram and P is the point of intersection of its diagonals. If O is any point, then overline(OA)+overline(OB)+overline(OC)+overline(OD)=

The positon vectors of points P, Q, R are given by overline(p)=overline(a)-2overline(b)+3overline(c), overline(q)=-2overline(a)+3overline(b)+2overline(c), overline(r)=-8overline(a)+13overline(b) . If the points P, Q, R are collinear, then the ratio in which point P divides the line segment RQ is

If overline(a)=hat(i)+3hat(j), overline(b)=2hat(i)+5hat(j), overline(c)=4hat(i)+2hat(j) and overline(c)=t_(1)overline(a)+t_(2)overline(b) , then

Given a cube ABCDA_1 B_1C_1D_1 with lower base ABCD and the upper baseA_1B_1C_1D_1 and the lateral edges A A_1, BB_1,, C C_1, DD_1. M and M_1, are the centres of the faces ABCD, and A_1 B_1 C_1 D_1 respectively. O is a point on the line MM_1. Such that overline(OA) + overline(OB) + overline(OC) + overline(OD) = overline(OM_1), . if overline(OM) = lambda overline(OM_1), then lambda is equal to

O is a point within the ∆ABC . P, Q , R are three points on OA , OB and OC respectively such that PQ||AB and QR||BC . Prove that RP||CA .

In triangleABC, if the points P, Q, R divides the sides BC, CA, AB in the ratio 1:4, 3:2, 3:7 respectively and the point S divides side AB in the ratio 1:3, then (overline(AP)+overline(BQ)+overline(CR)):(overline(CS)) =