in Fig . . The vector – B is added to vector A to get R = ( A – B ). The vector R = A + B is also shown in the same figure for comparison.
We can also use the parallelogram method to find the sum of two vectors. Suppose we have two vectors A and B . To add these vectors, we bring their tails to a common origin O as shown in Fig. .
(a). Then we draw a line from the head of A parallel to B and another line from the head of B parallel to A to complete a parallelogram O Q SP. Now we join the point of the intersection of these two lines to the origin O. The resultant vector R is directed from the common origin O along the diagonal (OS) of the parallelogram [Fig.
(c), the triangle law is used to obtain the resultant of A and B and we see that the two methods yield the same result. Thus, the two methods are equivalent. Example . Rain is falling vertically with a speed of m s – .
Winds starts blowing after sometime with a speed of m s – in east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella ? Fig. .
Answer The velocity of the rain and the wind are represented by the vectors v r and v w in Fig. . and are in the direction specified by the problem. Using the rule of vector addition, we see that the resultant of v r and v w is R as shown in the figure.
The magnitude of R is w m s m s The direction θ that R makes with the vertical is given by . w θ = θ = ° tan - . Therefore, the boy should hold his umbrella in the vertical plane at an angle of about o with the vertical towards the east.