IF `P_1, P_2, P_3, P_4` are points in a plane or space and `O` is the origin of vectors, show that `P_4` coincides with `Oiff( vec O P)_1+ vec P_1P_2+ vec P_2P_3+ vec P_3P_4= vec 0.`

IF P_(1),P_(2),P_(3),P_(4) are points in a plane or space and O is the origin of vectors,show that P_(4) coincides with Oiff vec OP_(1)+vec P_(1)P_(2)+vec P_(2)P_(3)+vec P_(3)P_(4)=vec 0

Given four points P_1,P_2,P_3a n dP_4 on the coordinate plane with origin O which satisfy the condition ( vec (O P))_(n-1)+( vec(O P))_(n+1)=3/2 vec (O P)_(n) . If P1 and P2 lie on the curve xy=1 , then prove that P3 does not lie on the curve

Given four points P_1,P_2,P_3a n dP_4 on the coordinate plane with origin O which satisfy the condition ( vec (O P))_(n-1)+( vec(O P))_(n+1)=3/2 vec (O P)_(n) . If P1 and P2 lie on the curve xy=1 , then prove that P3 does not lie on the curve

Given four points P_1,P_2,P_3a n dP_4 on the coordinate plane with origin O which satisfy the condition ( vec (O P))_(n-1)+( vec(O P))_(n+1)=3/2 vec (O P)_(n) (i) If P1 and P2 lie on the curve xy=1 , then prove that P3 does not lie on the curve (ii) If P1,P2,P3 lie on a circle x^2+y^2=1, then prove that P4 also lies on this circle.

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.

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.

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

If P_1:vec r.vecn_1-d_1=0 P_2:vec r.vec n_2-d_2=0 and P_3:vec r.vecn_3-d_3=0 are three planes and vec n_1, vec n_2 and vec n_3 are three non-coplanar vectors, then three lines P_1= 0 , P_2=0 ; P_2=0 , P_3=0 ; P_3=0 P_1=0 are a. parallel lines b. coplanar lines c. coincident lines d. concurrent lines

If P_1:vec r.vecn_1-d_1=0 P_2:vec r.vec n_2-d_2=0 and P_3:vec r.vecn_3-d_3=0 are three planes and vec n_1, vec n_2 and vec n_3 are three non-coplanar vectors, then three lines P_1= P_2=0 ; P_2= P_3=0 ; P_3= P_1=0 are a. parallel lines b. coplanar lines c. coincident lines d. concurrent lines

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