A plane body has perpendicular axes OX and OY marked on it and is acted on by following forces 5P in the direction OY, 4P in the direction OX, 10P in the direction OA where A is the point `(3a, 4a)` 15 P in the direction AB where B is the point `(-a, a)`.
Express each force in the vector form & calculate the magnitude & direction of the vector sum of these forces.

Answer

`5Phat(j), 4Phat(i), 6Phat(i)+8Phat(j), -12Phat(i)-9Phat(j), sqrt(20)P, tan^(-1)[-2]` with +ve x-axis

Explanation

`vecF_1=5Phatj,hatF_2=4P_hati,hatF_3=10P((3hati+4hatj)/5)=(6hati+8hatj)P`
`vecF_4=15P(((-hati-3hatj)+(hatj-4hatj)))/5=-12Phati-9Phatj`
`vecF=vecF_1+vecF_2+vecF_3+vecF_4=5Phatj+4Phati+6Phati+8hatj-12Phati-9Phati=-2Phati+4Phatj`
`|vecF|=Psqrt((-2)^2+(4)^2)=sqrt(20)P`
`tan alpha=4/(-2)implies alpha=tan^(-1)(-2)`