Question

Hello,

The question is the following: the resultant of two vectors, A and B, is a vector of 40 meters oriented towards the north. If the magnitude of A is 30 m and it forms an angle of 30 degrees towards the south in relation to the west, determine the vector of B. Thank you for your help.

Answer 1

Hello!

First, let's begin with drawing a vector diagram:

Since R is the result of vector sum of A and B (or algebraic sum of their components like A_S and A_W), and since R does not have neither W- nor E-component, we conclude that the horizontal component of B (BW or BW) must 'cancel' AW. It means that

$$A_W+B_x=0,\\ B_x=-A_W=B_E.$$

Draw this vector:

$$B_E=|A_W|=30\cos30°=26\text{ m toward East}.$$

Consider the vertical component of R. It is a result of subtracting the vertical component of A from the vertical component of B (because A points downward):

$$R=B_y-A_y,\\ A_y=A_S,\\ R=B_y-A_S,\\ B_y=R+A_S.\\ B_y=R+A\sin30°=40+30\sin30°=\\=55\text{ m toward North}.$$

Draw it:

Therefore, according to Pythagorean theorem, we have

$$B=\sqrt{B_E^2+B_N^2}=61\text{ m},\\\space\\ \theta=\arctan\frac{B_N}{B_E}=65°\text{ East of North}.$$