Vector a has a magnitude of 5.2 units and is directed east. Vector b has a magnitude of 4.3 units and is directed 35° west of north. By constructing vector diagrams, find the magnitudes and directions of (a) a + b, and (b) a-b.

The figure below shows the vector diagram, representing vectors and .

To add the vectors, we need to resolve vector into its components along the unit vectors and respectively. The unit vector points towards the north while the unit vector points toward the east direction (as can be seen from the figure above).
We assume that the angle subtended by vector is such that vector is degrees west of north. We also assume that the magnitude of vector be represented by b whereas the magnitude of vector is represented by a .
Therefore the vector component of vector towards the north, along the unit vector is given as whereas the vector component along the west, opposite to the direction of unit vector is (refer figure below).


Given:



Therefore one can write the vector as:

Where is the component of vector along the unit vector .
Since vector points in the east, along the unit vector , the value of component is equal to the magnitude of vector that is a .
Substituting the given values, we have
…… (1)
The vector is given as:



Substituting the given values, we have

…… (2)
The negative sign in the component of vector indicates that the component points opposite to the direction of unit vector that is along west.
(a) The addition of vector and vector is given as:
…… (3)
Since there is no component of vector along the unit vector , we have .

From equations (1) and (2), we have



Substituting the above values in equation (3), we have


Therefore the vector is .


Let us assume that the vector makes an angle with the horizontal. One can calculate the value of using the relation given below as:



Substituting the value of , , and from above, we have


The positive angle shows that the angle is measured counterclockwise with respect to horizontal.

Therefore vector makes an angle of with the horizontal.


The difference of vector and vector is given as:
…… (4)

Since there is no component of vector along the unit vector , we have .

From equations (1) and (2), we have



Substituting the given values in equation (4), we have



Therefore the vector is .


Let us assume that the vector makes an angle with the horizontal. One can calculate the value of using the relation given below as:







Substituting the value of , , and from above, we have


The negative sign above shows that the angle is measured clockwise with respect to the horizontal.

Therefore vector makes an angle of with the horizontal.

In such a problems, always proceed like as
1. Draw east, west and north-south on horizontal and vertical direction.
2. Resolve the vectors along horizontal and vertical direction.
3. According to the requirement of question, find the resultant .