`A B C D` are four points in a plane and `Q` is the point of intersection of the lines joining the mid-points of `A B` and `C D ; B C` and `A Ddot` Show that ` vec P A+ vec P B+ vec P C+ vec P D=4 vec P Q ,` where `P` is any point.

ABCD are four points in a plane and Q is the point of intersection of the lines joining the mid-points of AB and CD;BC and AD. Show that vec PA+vec PB+vec PC+vec PD=4vec PQ where P is any point.

If O is a point in space, A B C is a triangle and D , E , F are the mid-points of the sides B C ,C A and A B respectively of the triangle, prove that vec O A + vec O B+ vec O C= vec O D+ vec O E+ vec O Fdot

If vec P and vec Q are two vectors, then the value of (vec P + vec Q) xx (vec P - vec Q) is

If A( vec a),B( vec b)a n dC( vec c) are three non-collinear points and origin does not lie in the plane of the points A ,Ba n dC , then point P( vec p) in the plane of the A B C such that vector vec O P is _|_ to planeof A B C , show that vec O P=([ vec a vec b vec c]( vec axx vec b+ vec bxx vec c+ vec cxx vec a))/(4^2),w h e r e is the area of the A B Cdot

If vec p xxvec q=vec r and vec p*vec q=c, then vec q is

The line joining the points 6vec a-4vec b-5vec c,-4vec c and the line Joining the points -vec a-2vec b-3vec c,vec a+2vec b-5vec c intersect at

Find the equation of a line: passing through the point vec c ,parallel to the plane vec r*vec n=p and intersecting the line vec r=vec a+tvec b

p A N D q are any two points lying on the sides D C a n d A D respectively of a parallelogram A B C Ddot Show that a r( A P B)=a r ( B Q C)dot

In A A B C ,\ P\ a n d\ Q are respectively the mid-points of A B\ a n d\ B C and R is the mid-point of A Pdot Prove that: a r\ ( P B Q)=\ a r\ (\ A R C)