You are probably more familiar with scalars. Scalars and vectors are both physical and mathematical quantities.
A scalar can be completely specified by its magnitude, in other words, how big it is.
Common physical quantities that are scalars include, length,
mass, time, and derived qualities such as volume and energy.
A vector, on the other hand, needs to be specified not only by its magnitude, but also by its direction.
Common physical quantities that are vectors include force,
velocity,
acceleration and moment.
Note that in print, the vector is normally expressed by bold letters.
But in handwritings we normally write a vector with an arrow on top of the letter.
Like other vectors, a force can be represented by an arrow.
It can be fully characterized by its point of action, its sense of direction,
and its magnitude.
The direction of the force can be described by the angle made by its line of action
and a reference line. 
Sometimes you might see negative angles.
This is because by sign convention, positive angle represents counterclockwise rotation and 
negative angle represents clockwise rotation from the reference line.
To perform vector addition we need to follow the parallelogram law.
Let's say we have two arbitrary vectors A and B, and we want to find the sum,
in other words, the resultant vector, R, which equals to A plus B.
First step we need to join the tails of the two vectors together so that they are concurrent.
Then we construct a parallelogram using A and B as the two sides.
And then draw an arrow that starts from the tails of A and B and points to the other end.
This arrow is the resultant vector R, representing both the magnitude and direction of R.
As a simplification to the parallelogram law we can use the triangle rule instead.
Again we start with the two vectors A and B, and we want to find their sum, R.
Instead of joining the tails of the two vectors, we now join them in a head-to-tail fashion. As you can see,
now the head of vector A is connected with the tail of vector B.
Then the resultant vector R can simply be represented by a narrow that starts from the tail of vector A to the head of vector B.
What if we want to do vector subtraction?
For example, what is R prime that equals to A minus B?
Since we know subtraction can be considered as addition with a negative quantity, therefore we can first find the vector, negative B, 
which has the same magnitude, but opposite direction as vector B.
Then we can simply
add vector A and vector negative B together, using again either parallelogram law or triangle rule.
Let's look at this example.
For forces F_1 and F_2, their magnitudes and directions are both given and we need to 
determine the magnitude and direction of the resultant force of these two.
For this problem we are going to apply the triangle rule.
So we join the two forces in the head to tail fashion.
So this vector is the resultant force vector F_R, and now the three forces form a triangle and 
based on the geometry given in the problem statement, we can calculate that this angle is 75°.
Now this problem becomes solving a triangle.
As you recall from trigonometry, for a triangle with the sides of lengths a, b and c, and the corresponding angles capital A, B and C, 
to determine the relation of these parameters we have law of sines,
as well as law of cosines.
Based on the information we have we know that a is 100, b is a 60 and angle C is 75°.
We can start with this equation to calculate the side length c,
and then use this equation to calculate angle B. And here are the results. And from here we can also calculate the angle right here,  
theta equals to 45° minus the angle B 34.5°, to be 10.5°.
So we not only have determined the magnitude of the resultant force, to be 102 newton, but we also determined the angle made by the force 
with the horizontal line to be 10.5°. And this answers the question.
Let's look at another example. In this case we are given the magnitude and the direction of the resultant force, and we are also given the 
directions of its two component forces, and we need to determine the magnitudes of the two compoment forces.
For this problem we are going to first apply the parallelogram law and make a parallelogram based on the information given.
And now we can visualize the two component forces we need to solve for.
Then let's make the triangle again, calculate the third angle, and then apply law of sines directly,
to calculate the magnitudes of these two forces.
