A jogger travels a route that has two parts. The first is a displacement ->A of 2.20 km due south, and the second involves a displacement ->B that points due east. (a) The resultant displacement ->A + ->B has a magnitude of 3.81 km.
What is the magnitude of B?
km
What is the direction of A + B relative to due south?
° west of south or east of south?
(b) Suppose that A - B had a magnitude of 3.81 km. What then would be the magnitude of B and what is the direction of A - B relative to due south?

a) B = 3.11 km.  θ= 54.7º E of S

b) B = 3.11 km  θ= 54.7º W of S

Explanation:

a)

Since we know the value of the total displacement, and the value of the displacement A and its direction, we can find the magnitude of B just applying the Pythagorean Theorem, as follows:

       

Solving for B, the only unknown, we get:

       

Now, applying some simple trig, we can find the angle that (A+B) makes with the S axis, as follows:

       

Since it's a positive number, applying the convention that the positive angles are measured counterclockwise, this means that this angle is measured East of South.

b)

If the magnitude of A-B is the same than the one for A+B, this means that the magnitude of B remains the same, i.e. 3.11 Km.But if we do graphically A-B, as it is the same as adding A + (-B),        we find that the angle of A-B is different to the one in A+B, even the       magnitudes of both displacements are the same.In this case, B is a negative number, because it's a displacement due west.So, applying the same trig that for a) we can find the angle that (A-B) makes with the S Axis, as follows:

     

So, since it is negative, it's measured clockwise from the S axis, so it's 54.7º W of S.