vector addition

Vector addition is the process of summing vectors. It is analagous to adding together more than one matrix, each having one and only one row. To accomplish this, one must add corresponding individual corresponding components, placing the results in a new vector. For example, to add <a, b, c, d> to <a, f, g, h>, you add a to e, b to f, etc. and obtain <a+e, b+f, c+g, d+h>
as your resultant vector.

If you are unfamiliar with this mathematical notation of vectors, consult the "vector" node.

If you wish to assimilate the knowledge in this node fully, you will benefit from graph paper and an understanding of the knowledge ensconced here.

Draw two lines, one from (1,1) to (1,3) and one from (1,3) to (2,5). The vector which represents the first line is (0,2), and the one for the second is (1,2).

Now draw a line from (1,1) to (2,5). What's the vector representing that line? It's (1,4), which happens to be (1,2) + (0,2). This is very logical when you think about it - the vector from (1,1) to (2,5), which we call the resultant vector of the other two, represents the entire "journey" made by the other two vectors - they are merely components of it.

This relationship can come in very useful (admittedly, mainly in math classes). You can learn a lot about the vector triangles you've been drawing by using geometry and trigonometry. It's possible to calculate the resultant or either of the other two vectors by knowing the other two. If you know the resultant and one of the others, then simply subtracting the other from the resultant will give you the remaining vector.

Vectors are often used in velocity calculations. A system where velocity is at a constant, and then is effected and changed by some other factor is a consistent player in maths questions. The "riverboat question" comes up again and again.

In this scenario, a boat is said to be sailing through the water at a velocity of, say, (0,-3) - directly downwards. The velocity of the water is said to be, say, (-5,0) - directly right. What is the resultant velocity of the boat?

Easy. Resultant velocity = (0,-3) + (-5,0) = (-5,-3). We can use the techniques we learnt earlier to work out the magnitude of this resultant velocity vector, which is the speed -

             h2 = a2 + b2
             h2 = -52 + -32
             h = √34
             h = 5.8 km h-1

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.

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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.

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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.