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 P O+ vec O Q= vec Q O+ vec O R , show that the point, P ,Q ,R are collinear.

Write vec P Q+ vec R P+ vec Q R in the simplified form.

If vec p is a unit vector and ( vec x-\ vec p).( vec x+\ vec p)=80 , then find | vec x|

A B C D is quadrilateral such that vec A B= vec b , vec A D= vec d , vec A C=m vec b+p vec ddot Show that he area of the quadrilateral A B C Di s1/2|m+p|| vec bxx vec d|dot

If P ,Q and R are three collinear points such that vec P Q= vec a and vec Q R = vec bdot Find the vector vec P R .

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

Let vec a= hat i+2 hat ja n d vec b=2 hat i+ hat jdoti s |vec a|=| vec b|? Are the vectors vec a a n d vec b equal?

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(P) + vec(Q)| = |vec(P) - vec(Q)| , find the angle between vec(P) and vec(Q) .

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 :