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

If x ,1,a n dz are in A.P. and x ,2,a n dz are in G.P., then prove that x ,a n d4,z are in H.P.

Let vec r_1, vec r_2, vec r_3, .....vec r_n be the position vectors of points P_1,P_2, P_3 ,......P_n relative to the origin O . If the vector equation a_1 vec r_1+a_2 vec r_2+.....a_n vec r_n=0 hold, then a similar equation will also hold w.r.t. to any other origin provided a) a_1+a_2+......a_n=n b) a_1+a_2+....a_n=1 c) a_1+a_2+....a_n=0 d) a_1=a_2=a_3+a_n=0

If .^(2n+1)P_(n-1): .^(2n-1)P_(n)=3:5 , find n.

Let P M be the perpendicular from the point P(1,2,3) to the x-y plane. If vec O P makes an angle theta with the positive direction of the z- axis and vec O M makes an angle phi with the positive direction of x- axis, w h e r eO is the origin and thetaa n dphi are acute angels, then a. costhetacosphi=1//sqrt(14) b. sinthetasinphi=2//sqrt(14) c. ""tanphi=2 d. tantheta=sqrt(5)//3

If n be a positive interger and p_(n) denotes the product of the binomial coefficients in the expansion of (1+x)^(n)," Prove that, "(P_(n+1))/(P_(n))=(n+1)^(n)/(n!) .

The projection of point P( vec p) on the plane vec r. vec n=q is ( vec s) , then a. vec s=((q- vec p. vec n) vec n)/(| vec n|^2) b. vec s=p+((q- vec p. vec n) vec n)/(| vec n|^2) c. vec s=p-(( vec p. vec n) vec n)/(| vec n|^2) d. vec s=p-((q- vec p. vec n) vec n)/(| vec n|^2)

If the vectors 3 vec p+ vec q ;5p-3 vec qa n d2 vec p+ vec q ;4 vec p-2 vec q are pairs of mutually perpendicular vectors, then find the angle between vectors vec pa n d vec qdot

Given that A_1,A_2,A_3, A_n are n points in a plane whose coordinates are x_1,y_1),(x_2,y_2) ,(x_n ,y_n), respectively. A_1A_2 is bisected at the point P_1,P_1A_3 is divided in the ratio A :2 at P_2,P_2A_4 is divided in the ratio 1:3 at P_3,P_3A_5 is divided in the ratio 1:4 at P_4 , and so on until all n points are exhausted. Find the final point so obtained.

Distance of point P( vec p) from the plane vec rdot vec n=0 is a. | vec pdot vec n| b. (| vec pxx vec n|)/(| vec n|) c. (| vec pdot vec n|)/(| vec n|) d. none of these