# 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.

![Page 0 Blog Entry 12 1](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-1.jpg)

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:

![Page 0 Blog Entry 12 2](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-2.jpg)

![Page 0 Blog Entry 12 3](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-3.jpg)

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

![Page 0 Blog Entry 12 4](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-4.jpg)

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:
![Page 0 Blog Entry 12 5](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-5.jpg)

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.

![Page 0 Blog Entry 12 6](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-6.jpg)

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).

![Page 0 Blog Entry 12 7](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-7.jpg)

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

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

![Page 0 Blog Entry 12 8](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-8.jpg)

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

![Page 0 Blog Entry 12 9](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-9.jpg)

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:

![Page 0 Blog Entry 12 10](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-10.jpg)

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:

![Page 0 Blog Entry 12 11](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-11.jpg)

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:

![Page 0 Blog Entry 12 12](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-12.jpg)

For the above case, the following will be true:

![Page 0 Blog Entry 12 13](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-13.jpg)

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:

![Page 0 Blog Entry 12 14](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-14.jpg)

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:

![Page 0 Blog Entry 12 15](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-15.jpg)

**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:

![Page 0 Blog Entry 12 16](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-16.jpg)

Now I can find the components of each vector:

![Page 0 Blog Entry 12 17](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-17.jpg)

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:

![Page 0 Blog Entry 12 18](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-18.jpg)

If I sketch this, it would look like this:

![Page 0 Blog Entry 12 19](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-19.jpg)

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

![Page 0 Blog Entry 12 20](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-20.jpg)

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:

![Page 0 Blog Entry 12 21](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-21.jpg)

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

![Page 0 Blog Entry 12 22](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/09/page-0-blog-entry-12-22.jpg)

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.

1. #1 dhanashri
September 16, 2008

great teaching, pl. continue in this kind of simple, fluid lang with many more illustrations

2. #2 rhett
September 16, 2008

I will be happy to add more topics. Do you have any requests?

3. #3 Ramanathan
September 23, 2008

Can you publish some tutorial on relative velocity

4. #4 rhett
September 24, 2008
5. #5 John
October 5, 2008

did you not get 5.39 feet isn’t that more than 5 feet? LOL

6. #6 John
October 5, 2008

OOO never mind! IM STUPID LOL

7. #7 Barbara
October 20, 2008

This helps a lot. Thanks.

8. #8 VINITH. NAIR
October 30, 2008

Amazing tutorial for students preparing for the basics of the physics. Helped me a lot. thanxxx.

9. #9 Anonymous
November 5, 2008

hotdog

10. #10 Ibrahim Al-Aghbari
November 15, 2008

Great,

Hope it helps in my next test 😉

regards,
Ibrahim

11. #11 Alek
January 6, 2009

Hi. Good site.

12. #12 Maxwell
April 24, 2009

REALLY INFORMATIVE

COULD YOU TOUCH ON VECTOR REPRESENTATION
THANK YOU.

13. #13 clayne
June 30, 2009

tnx to this great teaching, this help us more to understand this about this. God bless you more

14. #14 sandrar
September 10, 2009

Hi! I was surfing and found your blog post… nice! I love your blog. Cheers! Sandra. R.

15. #15 angelina jolie
September 10, 2009

I love your site. Love design!!! I just came across your blog and wanted to say that I?ve really enjoyed browsing your blog posts. Sign: ndsam

16. #16 Wolf24
October 22, 2009

It is important to conduct research on this, and to make such research available to educational and professional practice communities in Scotland and beyond. ,

17. #17 Kelvin89
October 23, 2009

I can tell they are about to go to seed. ,

18. #18 RAR
October 25, 2009

@John
that thing’s supposed to be 4.39

19. #19 justin
January 8, 2010

Good site. The second picture is wrong though. It should be 3.61 feet not 5.29, in case that confused anyone. Otherwise very good.

20. #20 Eshan
February 4, 2010

very understandable and conceptual……would you please write on the projectiles in two dimensions….just explaining it from the vector concept and possible cases by using vector equation

21. #21 raheel khattak
March 15, 2010

iam not satisfied plz write to the point

22. #22 roza
November 25, 2010

thanx i understood so many things but i have a quastion
can i add two different vectors if they had the same quantity?
for example if they both were M or M/s2 or anything else??

23. #23 shakti
January 31, 2011

i am happy to get the basic information about vector
thanku very much

24. #24 HemKurs
February 9, 2011

Vector matematik geometri fizik en çok sevdiğim derslerdi dostum güzel paylaşım bulunmuşsun çok sağolus ne diyeyim thank dersler için bende burda hem halk eğitim kurslarına gidiyorum beklerim gelin sizsde eğğtimi kalitesini görün şaşıracaksiniz. hemkurs good..

25. #25 KolayPastalar
February 9, 2011

hi vectors very good Rhett Allain çok güzel bilgiler vermişysin bende geometrik düzenler bacek verelir üzerinde çalışmıştım tabi fırsat olmadıkı matematik doktorası yapmaya şimdi bir pastane açıp orad kendim pastalar yapıyorum kolay pastalar pratik bilgiler vererek isnasnların vakitlerini boşa harcamalarının önenems geçmiş oluyorum thank allain

26. #26 turk yesilcam
February 13, 2011

goo hic canım uyumak istemipyor hep turk yeşilçam seksilerinden sıcak viedohlarırın aklından geirire zerrin ablaya finge sarısını gözlerinmi önnde kalıdır turk sahnelri sevismeler süppeprlerdi good vectors hii

27. #27 Kishore Kumar Saxena
September 6, 2011

Thank you very much for the text but want know more about vectors and their uses in life would you please.

28. #28 Uche uzor
December 15, 2011

D topic has been interesting and i hv gotten enough knowledge but i will request if calculations are made big bcos i dont seem to see d calculations very clearly

29. #29 Uche uzor
December 15, 2011

D topic has been interesting and i hv gotten enough knowledge but i will request if calculations are made big bcos i dont seem to see d calculations very clearly

30. #30 jone alia
January 22, 2012

can u please brief about that how to determine whether x component to take with cos or y component with sin ?
and in the above example in the note u said that 6 feet are going against the triangle so u took x component with sin.. how to determine this .
thanks and good blog