A particle, in equilibrium, is subjected to four forces `vecF_1, vecF_2, vecF_3` and `vecF_4`, ` vec F_1 =-10 hat k , vec F _2 =u(4/13 hat i-12/13 hat j+3/13 hatk) , vec F _3 =v(-4/13 hat i-12/13 hat j+3/13 hatk), vec F_4 =w(cos theta hat i+sin theta hat j) ` then find the values of u,v and w

Explanation

Since, the particle is in equilibrium.
`F_(1)+F_(2)+F_(3)+F_(4)=0`
`-10hatk+u((4)/(13)hati-(12)/(13)hatj+(3)/(13)hatk)+v(-(4)/(13)hati-(12)/(13)hatj+(3)/(13)hatk)+w(costhetahati+sinthetahatj)=0`
`implies((4u)/(13)-(4v)/(13)+wcostheta)hati+((-12)/(13)u-(12)/(13)v+wsintheta)hatj+((3)/(13)u+(3)/(13)v-10)hatk=0`
`implies(4u)/(13)-(4v)/(13)+wcostheta=0` . . . (i)
`-(12)/(13)u-(12)/(13)v+wsintheta=0` . . . (ii)
`(3)/(13)u+(3)/(13)v-10=0`
From Eq. (iii), we get `u+v=(130)/(3)`
From eq. (ii), we get
`-(12)/(13)(u+v)+wsintheta=0`
`implies-(12)/(13)((130)/(3))+wsintheta=0`
`implies w=(40)/(sintheta)=40` cosec `theta`
On substituting the value of w in eqs. (i) and (ii), we get
`u-v=-130cot theta`
and `u+v=(130)/(3)`
On solving we get
`u+(65)/(3)-65 cot theta`
`v+(65)/(3)+65cot theta and w=40" cosec "theta`.