The lengths of two opposite edges of a tetrahedron are `a` and `b ;` the shortest distane between these edges is `d ,` and the angel between them is `theta` Prove using vectors that the volume of the tetrahedron is `(a b dsi ntheta)/6` .

If the angel between unit vectors vec aa n d vec b60^0 , then find the value of | vec a- vec b|dot

If theta is th angle between the unit vectors a and b , then prove that cos(theta/2)=1/2| vec a+ vec b| ,and sin(theta/2)=1/2| vec a- vec b|

Let A( vec a)a n dB( vec b) be points on two skew lines vec r= vec a+lambda vec pa n d vec r= vec b+u vec q and the shortest distance between the skew lines is 1, w h e r e vec pa n d vec q are unit vectors forming adjacent sides of a parallelogram enclosing an area of 1/2 units. If angle between A B and the line of shortest distance is 60^0, then A B= a. 1/2 b. 2 c. 1 d. lambda R={10}

The position vectors of the four angular points of a tetrahedron are A( hat j+2 hat k),B(3 hat i+ hat k), C(4 hat i+3 hat j+6 hat k)a n dD(2 hat i+3 hat j+2 hat k)dot Find the volume of the tetrahedron A B C Ddot

Define unit vector vec A and vec B are two vectors and theta is the angle between them. If |vecA xx vecB | = sqrt3(vecA.vecB) , Calculate the value of theta .

If a current passes through a metal conducting wire of area of cross section A, the drift velocity of free electrons inside the metal is v_d=1/( n e A) , where the amount of electric charge of an electron =e and the number of free electrons per unit volume of the metal=n. The applied electric field on the wire is E=V/l , where a potential difference V exists between two points, l apart, along the length of the wire. IF R is the resistance of the wire between those two points, then the resistivity of its material is rho=(RA)/l .Besides the mobility (mu) of the free electrons inside a wire is defined as their drift velocity for a unit applied electric field. Two copper wires have both lengths and radii in the ratio 1:2 if the ratio between the electric currents flowing through them is also 1:2 , what would be the ratio between the drift velocities of free electrons?

Express the relation between atomic radius (r) and edge length (a) in b.c.c. unit cell.

If O A B C is a tetrahedron where O is the origin and A ,B ,a n dC are the other three vertices with position vectors, vec a , vec b ,a n d vec c respectively, then prove that the centre of the sphere circumscribing the tetrahedron is given by position vector (a^2( vec bxx vec c)+b^2( vec cxx vec a)+c^2( vec axx vec b))/(2[ vec a vec b vec c]) .