Compute the resultant of three coplanar vectors P, 2P and 3P inclined at `120^@` with one another.

Explanation

Let the x-axis taken along vector P. Then the vectors 2P and 3P will [Fig.2.33] make angles `120^@` and `240^@` respectively with the x-axis, . Resolving the vectors into components along x and y axes and adding ,we get the sums `R_x and R_y` respectively along x and y axes.

`R_x=Pcos 0^@+2P cos 120^@+3P cos 240^@`
`therefore R_x=P+2P(-1/2)+3P(-1/2)=P-P-3/2P=-3/2P`
`R_y=P sin 0^@+2P sin 120^@+3P sin 240^@`
`=0+2Pxxsqrt(3)/2+3P+3P(-sqrt(3)/2)=(2-3)sqrt(3)/2P=-sqrt(3)/2 P`
Hence, the value of the resultant `vec (R)` is.
`R=sqrt(R_(x)^(2)+R_(y)^(2))=sqrt(9/4P^2+3/4P^2)=sqrt(3)P`.
`vec(R)` makes an angle `theta` with the x-axis such that,
`tan theta=(R_(y))/(R_(x))=(-sqrt3/2)/(-3/2P)=1/sqrt(3)= tan 30^@ or, tan 210^@`
As `R_x and R_y` are both negative, R must be in the third quandrant and `theta =210^@`.