Prove that the vector relation `pveca+qvecb +rvecc+….=0` will be inependent of the orign if and only if `p+q+r+.=0, where p,q,r………` are scalars.

Let veca, vecb and vecc be three non-coplanar unit vectors such that the angle between every pair of them is pi//3 . If veca xx vecb + vecb xx vecc =pveca + qvecb + rvecc , where p, q and r are scalars, then the value of (p^(2) + 2q^(2)+ r^(2))/q^(2) is:

If p^(q')=512 find the minimum possible value of (p) (q) (r) , where p, q, r are positive integers.

veca, vecb, vecc are non-zero unit vector inclined pairwise with the same angle theta . P,q,r are non-zero scalars satisfying veca xx vecb + vecb xx vecc=pveca + qvecb + rvecc . Now, answer the following questions: Volume of parallelogram with edges a,b and c is equal to:

If A,B,C are three points with position vectors veci+vecj,veci-hatj and pveci+qvecj+rveck respectiey then the points are collinear if (A) p=q=r=0 (B) p=qr=1 (C) p=q,r=0 (D) p=1,q=2,r=0

veca, vecb, vecc are non-zero unit vector inclined pairwise with the same angle theta . P,q,r are non-zero scalars satisfying veca xx vecb + vecb xx vecc=pveca + qvecb + rvecc . Now, answer the following questions: q/2+2 cos theta is equal to:

If lines px+qy+r=0,qx+ry+p=0 and rx+py+q=0 are concurrent,then prove that p+q+r=0 (where p,q,r are distinct )

IF the direction ratios of two vectors are connected by the relations p + q + r = 0 and p^(2) + q^(2) - r^(2) = 0. Find the angle between them.

Consider the following statements P: Suman is brilliant.Q: Suman is rich.R: Suman is honest.The negative of the statement. "Suman is brilliantand dishonest if and only if Suman is rich" can beexpressed as (a) ~(Q (0 P ~ R) (b) ~ Q R ^^ R (c) ~ (P ^^ ~ R) Q (d) ~ P ^^ (Q ~R)