Locus of the point P, for which `vec(OP)` represents a vector with direction cosine `cos alpha = (1)/(2)` (where O is the origin) is

A variable line through the point P(2,1) meets the axes at a an d b . Find the locus of the centroid of triangle O A B (where O is the origin).

A straight line L with nagative slope passes through the point (9,4) and cuts the positive coordinate axes at the points P and Q respectively . Minimum value of OP+OQ , as L varies (where O is the origin ), is-

A point P which represents a complex number z, moves such that |z-z_(1)|= |z-z_(2)|, then the locus of P is-

Let O be the origin and vec(OX) , vec(OY) , vec(OZ) be three unit vector in the directions of the sides vec(QR) , vec(RP),vec(PQ) respectively , of a triangle PQR. |vec(OX)xxvec(OY)|=

Let P and Q be any two points on the lines represented by 2x-3y = 0 and 2x + 3y = 0 respectively. If the area of triangle OPQ (where O is origin) is 5, then which of the following is not the possible equation of the locus of mid-point of PQ?

Show that the image of the point (h,k) with respect to the striaight line x cos alpha+ y sin alpha=p is the point (2 p cos alpha- h cos2 alpha- k sin 2 alpha, 2p sin alpha- h sin 2 alpha- k cos 2 alpha) .

Statement 1:Let A( vec a),B( vec b)a n dC( vec c) be three points such that vec a=2 hat i+ hat k , vec b=3 hat i- hat j+3 hat ka n d vec c=- hat i+7 hat j-5 hat kdot Then O A B C is a tetrahedron. Statement 2: Let A( vec a),B( vec b)a n dC( vec c) be three points such that vectors vec a , vec ba n d vec c are non-coplanar. Then O A B C is a tetrahedron where O is the origin.

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 2vec(AC)=3vec(CB) , then prove that 2vec(OA)+3vec(OB)=5vec(OC) where O is the origin.