let `baru and barv` be unit vectors. If `bar omega` is a vector such that `bar omega+(bar omega times bar u)=barv`. Then prove that the maximum volume of the parallelepiped formed by `baru,barv and bar omega` is `1//2`.

Let bar(PR)=3hati+hatj-2hatk and bar(SQ)=hati-3hatj-4hatk determine diagonals of a parallelogram PQRS and bar(PT)=hati+2hatj+3hatk be another vector. Then the volume of the parallelepiped determined by the vectors bar(PT),bar(PQ) and bar(PS) is

Position vector of a particle moving in x-y plane at time t is r=a(1- cos omega t)hat(i)+a sin omega t hat(j) . The path of the particle is

Let za n dw be two nonzero complex numbers such that |z|=|w|a n d arg(z)+a r g(w)=pidot Then prove that z=- bar w dot

IF bar(a),bar(b),bar(c ) are vectors from the origin to three points A,B,C show that bara times barb +barb times barc+barc times bara is perpendicular to the plane ABC.

Let bara,barb,barc be unit vectors such that bara+barb+barc=bar0 . Which one of the following is correct?

Given four non zero vectors bar a,bar b,bar c and bar d . The vectors bar a,bar b and bar c are coplanar but not collinear pair by pairand vector bar d is not coplanar with vectors bar a,bar b and bar c and hat (bar a bar b) = hat (bar b bar c) = pi/3,(bar d bar b)=beta ,If (bar d bar c)=cos^-1(mcos beta+ncos alpha) then m-n is :

If z=z_0+A( bar z -( bar z _0)), w h e r eA is a constant, then prove that locus of z is a straight line.

Let bar a,bar b,bar c be three non-coplanar vectors and bar d be a non-zero vector, which is perpendicularto bar a+bar b+bar c . Now, if bar d= (sin x)(bar a xx bar b) + (cos y)(bar b xx bar c) + 2(bar c xx bar a) then minimum value of x^2+y^2 is equal to

The complex number associated with the vertices A, B, C of DeltaABC are e^(i theta),omega,bar omega , respectively [ where omega,bar omega are the com plex cube roots of unity and cos theta > Re(omega) ], then the complex number of the point where angle bisector of A meets cumcircle of the triangle, is

Light wave travelling along y-direction. If the corresponding bar(E ) vector at any time along x-axis, the direction of bar(B ) vector at that time is along