Suppose that `vecp, vecq and vecr` are three non-coplanar vectors in `R^3`. Let the components of a vector `vecs` along `vecp, vecq and vecr` be 4,3 and 5 respectively. If the componenets of ths vector `vecs` along `(-vecp+vecq+vecr), (vecp-vecq+vecr) and (-vecp-vecq+vecr)` are x, y and z respectively, then the value of `2x+y+z` is.

If the vectors 3 vec p + vec q ; 5 vec p - 3 vecq and 2 vec p + vec q ; 4 vec p - 2 vec q are pairs of mutually perpendicular then sin( vec p , vec q) is :

If A (2,4) and B (-5,-3) are respectively the initial and final points of a vector vec v , find components of vec v and the magnitude of vec v .

If the two adjacent sides of two rectangles are represented by vectors vec p=5 vec a-3 vec b ; vec q=- vec a-2 vec b \ and \ vec r=-4 vec a- vec b ; vec s=- vec a+ vec b , respectively, then the angel between the vector vec x=1/3( vec p+ vec r+ vec s) \ and \ vec y=1/5( vec r+ vec s) is

P ,Q have position vectors vec a& vec b relative to the origin ' O^(prime)&X , Ya n d vec P Q internally and externally respectgively in the ratio 2:1 Vector vec X Y= 3/2( vec b- vec a) b. 4/3( vec a- vec b) c. 5/6( vec b- vec a) d. 4/3( vec b- vec a)

If vec alpha and vec beta are non-collinear vectors and vec a=(x+4y) vec alpha+ (2x+y+1) vec beta and vec b= (y-2x+2) vec alpha + (2x-3y-1) beta , find x and y so that 3 vec a= 2 vec b .

Find a vector vecr of magnitude 3sqrt2 units, which makes an angle of pi/4 and pi/2 with y and z-axes respectively.

Separation of mmotion of a systme of particles into motion of the centre of mass and motion about the centre of mass Show vec L = L = vec L + vecM R xx vecV where vec L = Sigma vecr_i xx vecp_i is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember vecr_i = vecr_i - vecR , rest f the notation is the standard notation used in the chapter. Note vecL and vecMR xx vecV can be said to be angular moemnta respectively about and of the centre of mass of the system of particles.

If vec a = x hat i +2 hat j - z hat k and vec b= 3 hat i- y hat j+hat k equal vectors, then find the value of x + y +z.

Let vec a , vec b and vec c be three non-zero vectors such that no two of them are collinear and ( vec axx vec b)xx vec c=1/3| vec b|| vec c| vec a . If theta is the angle between vectors vec b and vec c then a value of sintheta is : (1) (2sqrt(2))/3 (2) (-sqrt(2))/3 (3) 2/3 (4) (-2sqrt(3))/3