Basics: Vectors and Vector Addition

**pre-reqs:** trig

Think of the following two things. Temperature and wind speed. These are two different things that you could measure, but there is one big difference. Wind speed has two parts to it - how fast and which direction. Temperature is just one thing (no direction). Temperature is an example of a scalar quantity (just one piece of information). Wind speed is an example of a vector quantity - multiple pieces of information. Here are some other examples:


**Scalar:** mass, money, density, volume, resistance

**Vector:** velocity (most physicist reserve the word "speed" to mean just the magnitude), acceleration, force, momentum, displacement, electric field


Ok, I get it - but who cares? Well, if you are taking an introductory physics course, you should care. Here is a question I like to ask to start the discussion of vectors:


If I move 3 feet and then 2 feet, how far am I from where I started?


The answer is that there is no answer. I commonly get the quick answer of 5 feet, although this is only one possible answer. Let me illustrate this question with some pictures.


Here are 4 ways to add these two movements. Hopefully you can see from these examples that the answer will be somewhere between 1 and 5 feet. Try drawing a few combinations. Can you make one that is a total distance less than 1 foot? Can you make one more than 5 feet? No, you can't. But you can make anything in between these two. This is the most common mistake vector noobs make - they think they can treat vectors as though they were not vectors. Don't do that. Its bad.
**So then, how do you add vectors?**
In the above examples, some of them are not difficult to figure out. Actually, all but the last one is easy. **Note:** Here I am representing vectors by drawing arrows. In this representation, the length of the arrow represents how far I move and the direction of the arrow represents which direction. Convenient isn't it? Drawing arrows to represent vectors is conceptually useful, but actually not that practical (as you might see later). If the two movements are the same direction (or opposite direction) you could figure out how far you moved in your head - right? The other case that is reasonable is when the two movements are perpendicular to each other. In this case,the total distance is the hypotenuse of a right triangle. To find this, one can use the pythagorean theorem which says:



You have probably seen that before, yes? So for the case above, the distance is:


No problem, right? But what if the two vectors are not in the same direction and they are not perpendicular? Well, here is the key to vector addition: **Every vector can be broken into two vectors.** The same can be done with scalars, its just not usually very useful. For example, I can break up 3 as 1+2. I can break up 4 as -5+9 (why would I want to do that? maybe I have a good reason). Anyway, the same can be done with vectors, but it is important to remember that vectors are not scalars. To help with this distinction, I will write variables that represent vectors as different from variables that represent scalars. (All textbooks do this also). I will use an arrow above variables that are vectors, some textbooks write these variables with bold font (but that is not too helpful). So, I can write a vector:

I choose to break my random vector into two useful vectors, one pointing in the x-direction (whatever that is) and one pointing in the y-direction. This, in of itself, is not useful. If I also do it with other vectors, it will be useful. Imagine adding the following to vectors.


Looks complicated - yes? What if I break both vectors into vectors along the x- and y-axis (in this case I will say the x axis is horizontal and the y is vertical. It really doesn't matter which way your axes go as long as they are perpendicular and they don't change).


Here I am letting the vector A be broken into two vectors and doing the same thing for vector B.

**Vector Addition Commutes.**

Just like 3+4 = 4+3 = 7, the same is true for vectors:


This means that I can re arrange the vectors above and still add them: Here is my new picture:


Still busy, but perhaps now you can see the benefit. Now I have the two vectors in the x-direction added together and the two vectors and the y direction. The result of these two vectors are perpendicular. In essence, I have taken two vectors and broken them into 4. Here is the same thing written algebraically:


So, here is the strategy:
- Break vectors into x and y vectors
- Add the x vectors together (easy)
- Add the y vectors together (easy)
- Add the sum of x's to the sum of the y's (not bad using pythagorean)
- Done (well, done if you just want the distance) - more on this later.

**So, how do you find these "sub" vectors?**

Most textbooks call these sub-vectors vector components (what you break a vector into). It really is not too difficult to find them. Let's look at the vector **A** from above:


I added the angle that this vector is above the horizontal (or x-axis). When describing vectors, you need some way to describe which way they are pointing. For a 2-dimensional vector, one angle can do the job.

One of the great things about breaking a vector into components in the x- and y-direction is that these components are perpendicular. The components along with the original vector form a right triangle. Whenever you have a right triangle, you can use your right-triangle trig functions (sin cos etc..). **A note on trig functions:** There is really nothing too magical about these functions, they simply relate the sides of right triangles to the angle. Maybe I will write about this later. So, now that there is a right triangle, if I know the length of the hypotenuse and the angle ?, I can find the magnitude (length) of the two components. **Yet another note:** When writing just the magnitude (length) of a vector, it is a scalar quantity and thus does not need an arrow over it. A common representation for the magnitude of a vector is:


For the above case, the following will be true:


Please, please be careful. I have seen many many students think this is always the formula for finding the x- and y components. You have to look at your little picture of the right triangle. Sometimes it is backwards (just trust me and draw the picture). Also, it is possible for a component to be negative. The reason there can be negative components is because the scalar part is just a multiplier of a unit vector - huh? What does that mean?

**Unit vector:**
A unit vector has a length of one (with no units). The unit vector does have direction though. Here are two very useful unit vectors:


This shows two important unit vectors, one in the x-direction and one in the y-direction. Traditionally, unit vectors are represented with a "hat" over them instead of an arrow to denote their unit-vectorness. (some texts use i and j to represent the x and y unit vectors). Using these unit vectors helps one keep track of the direction of the components. This means that I can write the above example for vector **A** as:


**An Example:**

I think you are ready for a real example. Suppose I want you to move 3 meters at 25 degrees North of East and then 6 meters 40 degrees West of North. How far from the starting point would you have moved?
First, let me sketch this:


Now I can find the components of each vector:


Important things to note:
- for vector B, I calculated the x-component with the sin function. This is because if you look at the right triangle for this vector and its components, the vector component in the x-direction is the opposite side of the right triangle so that sin would be the appropriate function to use.
- For similar reasons the y-component uses the cos function
- The sign of the number in front of the x-hat vector is negative. I defined x-hat to be a vector pointing in the x-direction. The component for this vector points in the opposite direction thus it needs a negative sign. There are ways you can get this sign to come out automatically, but I recommend verifying the sign (make sure it is negative)
- Units are always important even though most physicists get lazy and leave them off (I am lazy also - but I put them on there because I care).
Now for the adding: Like before, I can re-arrange the order of the terms so that I get:


If I sketch this, it would look like this:


A right-triangle. The length of this hypotenuse would be:


This is the solution to the above problem, but what if I want to know the direction from the starting point to the finish point? Well, the angle of this vector above the x-axis would be:


or in the context of the question, 79 degrees North of West.
In reality,


this is the answer, just not in the same form. This component representation is actually (in my opinion) better and more useful than a magnitude and direction.

**More than two vectors:**

What if you need to add more than two vectors? Do the same thing as above.
- Sketch a picture
- Choose an x- and y-axis (this may not be obvious). If it is not obvious which direction to choose for the axes, pick whatever makes you happy. The x-and y- axes are not really so it doesn't matter.
- Break all the vectors into x- and y-components (be sure to use the correct trig function and be sure to verify the signs of the scalar components)
- Add up all the x-componets and then add up all the y-components
- Basically, that is the answer but you could use the pythagorean theorem to determine the length of the vector.
Remember it doesn't matter what kind of vectors these are.

**Subtraction:**
To subtract two vectors (say **A** - **B**), just multiply the components of vector B by a -1 and then add.

**The End:**

If you understand this, you are well on your way to become a vector-master (but there is much more to learn). The most important thing to remember is that with great power comes a greater responsibility to do good.