ABC is a triangle and P any point on BC. if ` vec P Q` is the sum of ` vec A P` + ` vec P B` +` vec P C` , show that ABPQ is a parallelogram and Q , therefore , is a fixed point.

A , B , C , D are any four points, prove that vec A B dot vec C D+ vec B Cdot vec A D+ vec C Adot vec B D=0.

Statement 1: let A( vec i+ vec j+ vec k)a n dB( vec i- vec j+ vec k) be two points. Then point P(2 vec i+3 vec j+ vec k) lies exterior to the sphere with A B as its diameter. Statement 2: If Aa n dB are any two points and P is a point in space such that vec P Adot vec P B >0 , then point P lies exterior to the sphere with A B as its diameter.

ABCD is a rectangle with A as the origin. vec b and vec d are the position vectors of B and D respectively. (i) What is the position vector of C? (ii) If P, Q, R and S are midpoints of sides of AB, BC, CD and DA respectively, find the position vector of P, Q, R,S,

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2 vec a + vec b) and (vec a - 3 vec b) externally in the ratio 1 :2. Also, show that P is the midpoint of the line segment RQ.

If vec p , vec q , vec r denote vector vec bxx vec c , vec cxx vec a , vec axx vec b , respectively, show that vec a is parallel to vec qxx vec r , vec b is parallel vec rxx vec p , vec c is parallel to vec pxx vec qdot

For any vectors vec a vec b , show that |vec a + vec b| le |vec a| + |vec b| .

If vec a , vec b , vec ca n d vec d are four vectors in three-dimensional space with the same initial point and such that 3 vec a-2 vec b+ vec c-2 vec d=0 , show that terminals A ,B ,Ca n d D of these vectors are coplanar. Find the point at which A Ca n dB D meet. Find the ratio in which P divides A Ca n dB Ddot

ABCD is a quadrilateral and E is the point of intersection of the lines joining the middle points of opposite side. Show that the resultant of vec (OA) , vec(OB) , vec(OC) and vec(OD) = 4 vec(OE) , where O is any point.

A B C D is a parallelogram. If La n dM are the mid-points of B Ca n dD C respectively, then express vec A La n d vec A M in terms of vec A Ba n d vec A D . Also, prove that vec A L+ vec A M=3/2 vec A Cdot

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