In a quadrilateral `P Q R S , vec P Q= vec a , vec Q R , vec b , vec S P= vec a- vec b ,M` is the midpoint of ` vec Q Ra n dX` is a point on `S M` such that `S X=4/5S Mdot` Prove that `P ,Xa n dR` are collinear.

if vec Ao + vec O B = vec B O + vec O C ,than prove that B is the midpoint of AC.

Find | vec a|a n d| vec b|,if( vec a+ vec b)dot( vec a- vec b) = 8 , | vec a|=8| vec b|dot

Given that vec adot vec b= vec adot vec c , vec axx vec b= vec axx vec ca n d vec a is not a zero vector. Show that vec b= vec cdot

If vec a= vec p+ vec q , vec pxx vec b=0a n d vec qdot vec b=0, then prove that ( vec bxx( vec axx vec b))/( vec bdot vec b)= vec qdot

If vec(PO) + vec(OQ) = vec(QO) + vec(OR) prove that the points P,Q,R are collinear .

Consider two points P and Q with position vectors vec(OP) = 3 vec a - 2 vec b and vec(OQ) = vec a + vec b .Find the position vector of a point R which divides the line joining P and Q in the ratio 2 : 1 (i) internally (ii) externally.

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.

If vec aa n d vec b are two vectors, then prove that ( vec axx vec b)^2=| vec adot vec a vec adot vec b vec bdot vec a vec bdot vec b| .

If vec a , vec b ,a n d vec c are three vectors such that vec axx vec b= vec c , vec bxx vec c= vec a , vec cxx vec a= vec b , then prove that | vec a|=| vec b|=| vec c|dot

If vectors vec a , vec b ,a n d vec c are coplanar, show that | vec a vec b vec c vec adot vec a vec adot vec b vec adot vec c vec bdot vec a vec bdot vec b vec bdot vec c|=0