`L_1a n dL_2` and two lines whose vector equations are `L_1: vec r=lambda((costheta+sqrt(3)) hat i+(sqrt(2)sintheta) hat j+(costheta-sqrt(3)) hat k)` `L_2: vec r=mu(a hat i+b hat j+c hat k)` , where `lambdaa n dmu` are scalars and `alpha` is the acute angel between `L_1a n dL_2dot` If the angel `alpha` is independent of `theta,` then the value of `alpha` is a. `pi/6` b. `pi/4` c. `pi/3` d. `pi/2`

Options

A. `(pi)/(6)`
B. `(pi)/(4)`
C. `(pi)/(3)`
D. `(pi)/(2)`

Answer

a

Explanation

Both the lines pass through the origin. Line `L_(1)` is parallel to the vector `vec(V_1)`
`" "vec(V_1)= (costheta+sqrt(3))hati+ (sqrt2 sin theta)hatj + (costheta-sqrt3)hatk`
and `L_(2)` is parallel to the vector `vec(V_2)`
`" "vec(V_2) = ahati+bhatj+chatk`
`therefore " "cosalpha= (vec(V_1)*vec(V_2))/(|vec(V_1)||vec(V_2)|)`
`= (a(costheta+ sqrt3)+ (bsqrt2)sintheta+c(costheta-sqrt3))/(sqrt(a^(2)+b^(2)+c^(2))sqrt((costheta+sqrt3)^(2)+ 2sin^(2)theta+ (costheta-sqrt3)^(2)))`
`((a+c)costheta+bsqrt2sintheta+ (a-c)sqrt3)/(sqrt(a^(2)+b^(2)+c^(2))sqrt(2+6))`
For `cos alpha` to be independent of `theta`, we get
`" "a+c=0 and b=0`
`therefore " "cosalpha = (2asqrt3)/(asqrt2 2sqrt2)= (sqrt3)/(2)`
or `" "alpha= (pi)/(6)`