`A(p,1,-1), B (2p,0,2) and C(2+2p,p,p)` are three points . Find the value of p so that the vectors `vec(AB)` and `vec(BC)` are perpendicular to each other .

Find the values of p so that the equation 2cos^2x-(p+3)cosx+2(p-1)=0 has a real solution.

The points (p+1,1), (2p+1,3) and (2p+2, 2p) are collinear, if

If A(1,p^2),B(0,1) and C(p ,0) are the coordinates of three points, then the value of p for which the area of triangle A B C is the minimum is (a) 1/(sqrt(3)) (b) -1/(sqrt(3)) (c) 1/(sqrt(2)) (d) none of these

If a/2=b/3 =c/4 =(3a-ab+4c)/p , then find the value of p.

If P(A)=2/3,P(B)=1/2 and P(AnnB)=1/6 then find the value of P(AnnB^C) .

P(1,0,-1), Q(2,0,-3),R(-1,2,0)a n dS(3,-2,-1), then find the projection length of vec P Qon vec R Sdot

Suppose that vec(p), vec(q) and vec(r) are three non-coplanar vectors in R. Let the components of a vector vec(s) along vec(p), vec(q) and vec(r) be 4, 3 and 5 respectively. If the components of this vector vec(s) " along" (-vec(p) + vec(q) + vec(r)), (vec(p) - vec(q) + vec(r)) and (-vec(p) - vec(q) + vec(r)) are x,y and z respectively, then the value of 2x + y + z is

If P(A) = 2/3, P(B) = 1/2 and P(A uu B) = 3/4 then find the value of P(A^C uu B^C) .

If P(AnnB)=1/2,P( barA nn barB )=1/3, P(A)=p , P(B)=2p , then find the value of pdot

Let vec pa n d vec q be any two orthogonal vectors of equal magnitude 4 each. Let vec a , vec b ,a n d vec c be any three vectors of lengths 7sqrt(15)a n d2sqrt(33), mutually perpendicular to each other. Then find the distance of the vector ( vec adot vec p) vec p+( vec adot vec q) vec q+( vec adot( vec pxx vec q))( vec pxx vec q)+( vec bdot vec p) vec p( vec bdot vec q) vec q+( vec bdot( vec pxx vec q))( vec pxx vec q)+( vec cdot vec p) vec p+( vec cdot vec q) vec q+( vec cdot( vec pxx vec q))( vec pxx vec q) from the origin.