1.

L_(1)andL_(2) are two lines whose vector equations are L_(1):vecr=lamda((costheta+sqrt3)hati+(sqrt2sintheta)hatj+(costheta-sqrt3)hatk) L_(2):vecr=mu(ahati+bhatj+chatk), where lamdaandmu are scalars and alpha is the acute angle between L_(1)andL_(2).If the anglealpha is independent of theta then the value of alpha is

Answer»

`(pi)/(6)`
`(pi)/(4)`
`(pi)/(3)`
`(pi)/(2)`

Solution :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)`


Discussion

No Comment Found

Related InterviewSolutions