Find the work done if a particle is displaced from `J(1m,2m,3m)` to `K(2m,3m,4m)` under a force (a) `vecF=(2hati+5hatj+6hatk)N` and (b) `vecF=(2xhati+3y^2hatj+4z^3hatk)N`

Explanation

(a) force is constant,
Position vector of J, `vecr_1=hati+2hatj+3hatk`
Position vector of K, `vecr_2=2hati+3hatj+4hatk`
displacement vector `vecd=vecr_2=hati+hatj+hatkm`
`W=vecF.vecd=(2hati+5hatj+6hatk).(hati+hatj+hatk)`
`=2+5+6=13J`
(b) `vecF=2xhati+3y^2hatj+4z^3hatk=F_xhati+F_yhatj+F_zhatk` force is
variable .
`W=intvecF.dvecr`
`=int_(x1)^(x2)F_(x)dx+int(y1)^(y2)F_ydy+int_(z1)^(z2)F_zdz`
`=int_(1)^(2)2xdx+int_(2)^(3)3y^2dy+int_(3)^(4)4z^3dz`
`|x^2|_1^2+|y^3|_2^3+|z^4|_3^4`
`={(2)^2-(1)^2}+{(3)^2-(2)^2}+{(4)^4-(3)^4}`
`=(4-1)+(27-8)+(256-81)`
`=3+19+175=197J`