`vec(a) "and" vec(b)` are two given mutually perpendicular vectors such that `|vec(a)|=4 "and" |vec(b)| =3. vec(c )` is a unit vector making an angle of `(pi)/(4)` and `(pi)/(3)` with `vec(a) "and" vec(b)` respectively. If (x,y) represents the number of possible vector `vec(c)` and the volume of tetrahedron formed by vectors `vec(a), vec(b),vec(c)` as adjacent sides respectively, then (x,y) is

Options

A. `(1,6)`
B. `(2,6)`
C. `(1,1)`
D. `(2,1)`

Answer

D

Explanation

Let `vec(c )=lambdavec(a)+muvec(b)+v(vec(a)xxvec(b))`. Then
` vec(a).vec(c )=vec(lambda) vec(a).vec(a) implies lambda=(1)/(4sqrt(2))`
`implies vec(b).vec(c)=mu vec(b).vec(b)`
`therefore mu=(1)/(6) and (vec(a)xxvec(b)).vec(c) = v(vec(a)xxvec(b))^(2)`
`therefore v=(v)/(144)'where V=(vec(a)xxvec(b)).vec(c)`.
Also `vec(c).vec(c) =lambdavec(a).vec(c)+mu vec(b).vec(c)+vV`
`1=lambda2sqrt(2)+mu.(3)/(2)+v(144v)`
`1=(1)/(2)+(1)/(4)+144 v^(2)`
v=pm(1)/(24) so,v=144xx(1)/(24)=6`