Prove that the necessary and sufficient condition for any four points in three-dimensional space to be coplanar is that there exists a liner relation connecting their position vectors such that the algebraic sum of the coefficients (not all zero) in it is zero.

For any four vectors veca, vecb, vec c , vecd (all non coplanar) any vector can be written as a linear combination of other three vectors.

A body is in translational equilibrium under the sctin of coplanar forces. If the torque of these force is zero about a point is it necessary that it will also be zero abut any other point?

Prove that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

A circle and a parabola y^2=4a x intersect at four points. Show that the algebraic sum of the ordinates of the four points is zero. Also show that the line joining one pair of these four points is equally inclined to the axis.

If vec a , vec b , vec ca n d vec d are four vectors in three-dimensional space with the same initial point and such that 3 vec a-2 vec b+ vec c-2 vec d=0 , show that terminals A ,B ,Ca n d D of these vectors are coplanar. Find the point at which A Ca n dB D meet. Find the ratio in which P divides A Ca n dB Ddot

If the resultant torque of all the forces acting on a body is zero about a point is it necessary that it will be zero about any other point?

Answer the following as true or false. (i) A zero vector cannot be assigned a definite direction. (ii) A zero vector may be regarded as having any direction. (iii) Two vectors are equal if they have same magnitude and direction regardless of the position of their initial points. (iv) The length of a unit vector is 2.

In an experiment on photoelectric effect, the emitter and the collector plates are placed at a separation of 10cm and are connected through an ammeter without any cell. A magnetic field B exists parallel to the plates. The work function of the emitter is 2.39 e V and the light incident on it has wavelengths between 400nm and 600nm. Find the minimum value of B for which the current registered by the ammeter is zero. Neglect any effect of space charge.

If veca, vecb and vecc are three non-zero, non-coplanar vectors,then find the linear relation between the following four vectors : veca-2vecb+3vecc, 2veca-3vecb+4vecc, 3veca-4vecb+ 5vecc, 7veca-11vecb+15vecc .

Statement 1: Let vec a , vec b , vec ca n d vec d be the position vectors of four points A ,B ,Ca n dD and 3 vec a-2 vec b+5 vec c-6 vec d=0. Then points A ,B ,C ,a n dD are coplanar. Statement 2: Three non-zero, linearly dependent coinitial vector ( vec P Q , vec P Ra n d vec P S) are coplanar. Then vec P Q=lambda vec P R+mu vec P S ,w h e r elambdaa n dmu are scalars.