Basic Vector Addition and Subtraction
Often, we see problems in Physics that ask you to compute the sum of two vectors, given the position, direction, and scalar. This is known as vector addition. Consider the following question as an example:
Vector A = 40cm, 60°
Vector B = 20cm, 90°
What is the sum of Vector A and Vector B?
To do this addition, realize that when two vectors are added like this, they simply form a new vector. We will call this Vector C.
Doing it the Visual Way
This is a method to add vectors visually, using measuring tools such as rulers and protractors.
Imagine Vector A from tail to head and plot it on a Cartesian Coordinate System like so:
& = head of vector
(y)
^

 &  Vector A 40cm, 60°
 /
 /
/
+> (x)



v
Now plot Vector B, except have the tail of Vector B
start at the
head of Vector A. Keep in mind the appropriate angles, it's the same as if you were at the
origin of the
Cartesian plane.
(y)
^


 &  Vector B. 20cm, 90°
 
 
 &  Vector A 40cm, 60°
 /
 /
/
+> (x)



v
Now you have both vectors in the correct position. To find Vector C, simply draw a line from the tail of Vector A to the head of Vector B. Use whatever tools of measurement you have to find the angle and length of the new vector. Remember to keep in mind the scale of the vector.
To subtract Vectors, we will use the same question as before in the example.
Vector A = 40cm, 60°
Vector B = 20cm, 90°
What is Vector A  Vector B?
& = head of vector
(y)
^

 &  Vector A 40cm, 60°
 /
 /
/
+> (x)



v
This time, for Vector B, we draw the Vector B in the
opposite direction from the tail of Vector A, so it would look like this:
(y)
^
 &  Vector A 40cm, 60°
 /
 / 
/ &  Vector B. 20cm, 270°
+> (x)



v
To find Vector C, draw a line from the tail of Vector A to the head of Vector B...just like addition, and measure.
To add more than two vectors, use the same method. Such as Vector A + Vector B + Vector C. Simply put the tail of B onto the head of A, and then put the tail of C onto the head of B.
Note: the addition and subtraction of number is commutative. So is the addition and subtractions of Vectors. You can add them any way you like, but the final vector will always be the same.
Doing it the Mathematical Way
There is an equation to find the sum of Vectors being added or subtracted. All it requires is some skill in Algebra and depending on the angles and lengths given, possibly a calculator with Cosine, Tangent, and Square Root functions.
Vector A = 40cm, 60°
Vector B = 20cm, 90°
What is the sum of Vector A and Vector B?
We did this problem visually before, but there is always error due to incorrect measurements or
misjudgement by the human eye. Now we will plug these values into an
equation and recieve an exact answer.
Mathematical Method of Vector Addition:
Vector A = Vector A + Vector B
Α = Length of Vector A
α = Angle of Vector A
Β = Length of Vector B
β = Angle of Vector B
Y coordinate of Vector C = Αsin(α) + Βsin(β)
X coordinate of Vector C = Αcos(α) + Βcos(β)
So we plug in:
Α = 40 cm
α = 60°
Β = 20 cm
β = 90°
Y coordinate of Vector C = 40sin(60°) + 20sin(90°)
= 40(0.866) + 20(1)
= 34.640 + 20.000
= 54.640 cm
X coordinate of Vector C = 40cos(60°) + 20cos(90°)
= 40(0.500) + 20(0)
= 20.000 + 0
= 20.000 cm
Now that we know the X and Y of Vector C, we can easily find out the
Hypotenuse, which will be the scalar of C by using the
Pythagorean Theorem.
length of vector C = (x^2 + y^2)^(1/2)
= (2985.530+400)^(1/2)
= (3385.530)^(1/2)
= 58.185 cm
We know that tan(C) will is x/y, so we plug in the x and y values for the
inverse tangent:
Angle of Vector C = tan^{1}(y/x)
= tan^{1}(54.640/20)
= tan^{1}(2.732)
= 69.896°
Now that we have the length (58.185) and the angle (69.896°) of Vector C, we have solved the addition of Vector A and Vector B accurately to three decimal places.
To subtract Vectors, just change the + signs in the original equations to  signs.
This too can be done commutatively and with more than two vectors.
Note 2: Remember to set your calculator to "degrees" not "radians" otherwise you will get strange answers."
Note 3: If you notice any errors in my arithmetic, please inform me. Thanks.