Find the resultant of the three displacement vectors in Fig. 1.16 by means of the component method.

Question

Find the resultant of the three displacement vectors in Fig. by means of the component method. The magnitudes of the vectors are A = 5.00 m, B = 5.00 m and C = 4.00 m.

Answer 1

First find the individual components of each of the vectors. Note, the angles given in the figure are measured in different ways so we have to think about the signs of the components. Here, the x component of vector A is negative and the y component of vector C (which is all it’s got!) is also negative. 

Using a little trig, the components of the vectors are:

and C_x = 0 C_y = −4.00 m

The resultant (sum) of all three vectors (which we call R) then has components

This gives the components of R. The magnitude of R is

If the direction of R (as measured from the +x axis) is θ, then

and naively pushing the tan−1 key on the calculator would have you believe that θ = −42.9°. Such vector would lie in the “fourth quadrant” as we usually call it. But we have found that the x component of R is negative while the y component is positive and such a vector must lie in the “second quadrant”, as shown in Fig. What has happened is that the calculator returns an angle that is wrong by 180° so we need to add 180° to the naive angle to get the correct angle. So the direction of R is really given by