Find the moment about the point `hat i+ 2hat j+ 3hat k` of a force represented by `hat i+hat j+hat k` acting through the point `2hat i+ 3hat j+hat k`.

Options

A. `2hat(i)+hat(j)+2hat(k)`
B. `hat(i)-hat(j)+3hat(k)`
C. `3hat(i)+2hat(j)-hat(k)`
D. `3hat(i)+hat(j)-4hat(k)`

Answer

D

Explanation

Force, `vec(F)` is given by `vec(F)=hat(i)+hat(j)+hat(k)`
`vec(OA)=hat(i)+2hat(j)+3hat(k) and vec(OB)=-2hat(i)+3hat(j)+hat(k)`
`:. vec(r)=vec(AB)=-2hat(i)+3hat(j)+hat(k)-hat(i)-2hat(j)-3hat(k)`
`=-3hat(i)+hat(j)-2hat(k)`
Moment `vec(M)` about the point `hat(i)+2hat(j)+3hat(k)`
`vec(M)=vec(r)xxvec(F)=(-3hat(i)+hat(j)-2hat(k))xx(hat(i)+hat(j)+hat(k))`
`|(hat(i) ,hat(j),hat(k)),(-3,1,-2),(1,1,1)|=hat(i)(1+2)-hat(j)(-3+2)+hat(k)(-3-1)`
`=3hat(i)+hat(j)-4hat(k)`