A bouy is attached to three tugboats by three ropes. The tugboats are engaged in a tug-0f-war. One tugboat pulls west on the buoy with a force `vec(F)_(1)` of magnitude `1000N`. The second tugboat pulls south on the buoy with a force `vec(F)_(2)` of magnitude `2000N`. The third tugboat pulls northeast (that is, half way between north and east), with a force `vec(F)_(3)` of magnitude `2000N`
a. Express each force in unit vector from `(hat(i),hat(j))`.
b. Calculate the magnitude of the resultant force.

Explanation

(a)
(b)
`vec(F)_(1)=-1000 hat(i) (N)`,
`vec(F)_(2)=-2000hat(j)(N)`,
`vec(F)_(3)=2000(cos45^(@)hat(i)+sin 45^(@)hat(j)(N)=1000sqrt(2)hat(i)+1000sqrt(2)hat(j)`
(c) `vec(F)_("resul tan t")=vec(F)_(1)+vec(F)_(2)+vec(F)_(3)=1000(sqrt(2)-1)hat(i)-1000(2-sqrt(2))hat(j) N`
`F_(x)=1000(sqrt(2)-1) N`
`F_(y)=-1000(2-sqrt(2))N`
`|vec(F)|=sqrt(F_(x)^(2)+F_(y)^(2))=1000(sqrt(2)-1)xxsqrt(3)=1000(sqrt(2)-1)sqrt(3) N`