The position vectors of four points P,Q,R annd S are 2a+4c,5a+`3sqrt(3)b+4c,-2sqrt(3)b+c and 2a+c` respectively, prove that PQ is parallel to RS.

If the position vectors of the points A,B and C be a,b and 3a-2b respectively, then prove that the points A,B and C are collinear.

If the position vectors of the points A , B , C ,Da r e2 hat i+4 hat k ,5 hat i+3sqrt(3) hat j+4 hat k ,-2sqrt(3) hat j+ hat ka n d2 hat i+ hat k respectively, prove that C D is parallel to A Ba n dC D=2/3A Bdot

If the position vectors of the points A(3,4),B(5,-6) and (4,-1) are vec a , vec b , vec c respectively compute vec a+2 vec b-3 vec c

If the position vectors of the points A(3,4),B(5,-6) and C (4,-1) are vec a , vec b , vec c respectively compute vec a+2 vec b-3 vec c

Factorize: 2sqrt(2)a^3+3sqrt(3)b^3+c^3-3sqrt(6)a b c

Let A,B,C be vertices of a triangle ABC in which B is taken as origin of reference and position vectors of A and C are veca and vecc respectively. A line AR parallel to BC is drawn from A PR (P is the mid point of AB) meets AC and Q and area of triangle ACR is 2 times area of triangle ABC Positon vector of Q for position vector of R in (1) is (A) (2veca+3vecc)/5 (B) (3veca+2vecc)/5 (C) (veca+2vecc)/5 (D) none of these

A B C D is a rhombus and P ,\ Q ,\ R ,\ S are the mid-points of A B ,\ B C ,\ C D ,\ D A respectively. Prove that P Q R S is a rectangle.

If the position vector of these points are vec(a) -2vec(b)+3 vec(c ), 2 vec(a)+3vec(b)-4 vec( c) ,-7 vec(b) + 10 vec(c ) , then the three points are

The perpendicular distance of the point P(3,3,4) from the x-axis is 3sqrt(2) b. 5 c. 3 d. 4

The length of the perpendicular drawn from the point P(3,4,5) on y-axis is a. 3sqrt(2) b. 5 c. sqrt(113) d. 5sqrt(2) e. sqrt34