Q. 55.0( 2 Votes )

Show that the fou

Answer :

Given the position vectors of points P, Q, R and S are,, and respectively.


We have


Rearranging the terms in the above equation,



Observe that the sum of coefficients on the LHS of this equation (5 + 6 = 11) is equal to that on the RHS (2 + 9 = 11).


We now divide the equation with 11 on both sides.




Now, consider the LHS of this equation.


Let , the position vector of some point X.



Recall the position vector of point P which divides AB, the line joining points A and B with position vectors and respectively, internally in the ratio m : n is



Here, m = 5 and n = 6


So, X divides RP internally in the ratio 5:6.


Similarly, considering the RHS of this equation, we have the same point X dividing SQ in the ratio 2:9.


So, the point X lies on both the line segments PR and QS making it the point of intersection of PR and QS.


As PR and QS are two straight lines having a common point, we have all the points P, Q, R and S lying in the same plane.


Thus, the points P, Q, R and S are coplanar and in addition, the position vector of the point of intersection of line segments PR and QS is or.


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