Four points P,Q,R and S with respective position vectors `vec p`,`vec q`,`vec r` and `vec s` are such that `5 vec p - 2vec q+ 6 vec r - 9 vec s= vec 0`. Show that the four points are coplanar and find the P.V. of the point in which the lines PQ and RS intersect.

A,B,C and D are four points with position vectors vec a , vec b , vec c and vec d respectively such that 5 vec a - 2 vec b + 6 vec c- 9 vec d= vec 0 . Show that the point A,B,C,D are coplanar and find find the P.V. of the point in which the lines AC and BD intersects.

Show that the four points P,Q,R,S with position vectors vecp,vecq,vecr,vecs respectively such that 5vecp-2vecq+6vecr-9vecs=vec0 , are coplanar. Also find the position vector of the point of intersection of the lines PR and QS.

If A,B,C,D are the points with position vectors vec a , vec b , 3 vec a+ 2 vec b and vec a- 2 vec b respectively , show that vec AC = 2 vec a + 2 vec b and vec DB= 3 vec b - a .

If vec r* vec a =0 = vec r* vec b ,where vec a and vec b are non-coplanar vectors then

If vec a , vec b , vec c are three non-coplanar vectors and vec d* vec a = vec d* vec b= vec d* vec c= 0 then show that vec d is zero vector.

If the vectors 3 vec p + vec q ; 5 vec p - 3 vecq and 2 vec p + vec q ; 4 vec p - 2 vec q are pairs of mutually perpendicular then sin( vec p , vec q) is :

Statement 1: if three points P ,Qa n dR have position vectors vec a , vec b ,a n d vec c , respectively, and 2 vec a+3 vec b-5 vec c=0, then the points P ,Q ,a n dR must be collinear. Statement 2: If for three points A ,B ,a n dC , vec A B=lambda vec A C , then points A ,B ,a n dC must be collinear.

Show that the points with position vectors vec a- 2 vec b+ 3 vec c , -2 vec a + 3 vec b+ 2 vec c and -8 vec a + 13 vec b are collinear, whatever vec a , vec b and vec c may be.