find the direction and magnitude of the resultant of the following velocities south, velocities impressed upon the particle:8m/s to south,12m/s to east and 3root2 m/s to north east.also give the direction of the resultant vector.

To sum the vectors it is advisable to take appropriate coordinate system and then finding the components of the vectors. Then after we can add the corresponding components just like scalars.

In the coordinate system , the vectors are summed as

(12,0)+(0,-8)+(3,3)=(15,-5) in m/s.

The resultant vector is 15i^-5j^ in m/s. This resultant vector makes an angle (alpha) with east direction ,where (alpha)= tan^-1(-1/3). Or (alpha)=18.43 degree east south . The magnitude is ( 15^2+5^2)^1/2=(250)^1/2=5 sqrt (10).m/s.