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## Linear algebra

### Unit 1: Lesson 1

Vectors- Vector intro for linear algebra
- Real coordinate spaces
- Adding vectors algebraically & graphically
- Multiplying a vector by a scalar
- Vector examples
- Scalar multiplication
- Unit vectors intro
- Unit vectors
- Add vectors
- Add vectors: magnitude & direction to component
- Parametric representations of lines

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# Unit vectors intro

Unit vectors are vectors whose magnitude is exactly 1 unit. They are very useful for different reasons. Specifically, the unit vectors [0,1] and [1,0] can form together any other vector. Created by Sal Khan.

## Want to join the conversation?

- So basically a unit vector is a vector which has only a unit value, as in, either in the horizontal or vertical dimension (Please correct me if I'm wrong). So what's the significance, or what's the point of learning about unit vectors? I mean, usual vectors are just scaled-up unit vectors. So unit vectors are just used for different representations?(6 votes)
- What is the significance? There are quite a few times we would like to have a unit vector. For instance, I recently made a game on the computer science section of the site. In the game, the player is allowed to shoot bullets. When the player clicks the screen, a bullet is fired from the player in the direction of where the mouse was clicked - but how do I make sure that the bullet travels at the same speed, no matter where I click? The solution is to first construct a unit vector from the position of the player to the point that was clicked, and then scale this unit vector to the desired length (which will be the velocity of the bullet). I used a lot of vectors to make this game. Here is the link:

https://www.khanacademy.org/computer-programming/creeper-invasion-v13/6444869583241216(36 votes)

- I frequently get mixed up between i and j. Is there any trick for remembering which one is which?(2 votes)
- they are in alphabetical order: both x,y,z and corresponding unit vectors i,j,k ;)(22 votes)

- Is there a name for the i and j with the hat on top ?(18 votes)
- You can also say, "In the i direction/ in the j direction." The reason we use unit vectors in the first place is to figure out a representation of the direction of the vector, so it makes sense to say it that way.(3 votes)

- What are the advantages of using these unit vectors notation?(2 votes)
- It makes it easy to add vectors. They're extensively used in Physics, engineering etc.(32 votes)

- Why do we use i and j as unit vectors, not x and y or a and b?(8 votes)
- It is notation. If you used x and y then somebody might think you're talking about a point or a line. These both have position. Vectors have no position, so they are distinguished from lines by i and j.(20 votes)

- So, a unit vector and a unit circle are related in some respect - one has a magnitude of 1, the other has a radius of one, right?(5 votes)
- Yes, all unit vectors will touch the unit circle when placed on the origin.(6 votes)

- What is the difference between unit vectors and basis vectors?(4 votes)
- A unit vector is a vector with length/magnitude 1.

A basis is a set of vectors that span the vector space, and the set of vectors are linearly independent. A basis vector is thus a vector in a basis, and it doesn't need to have length 1.(7 votes)

- Do you use vector notation in your life?(3 votes)
- It's quite rare that a day passes without giving me an opportunity to frame a problem/question/report in terms of vectors. If you've used a computer (since your post was likely made via a computer, I imagine that you have) you've had a use for vectors. Videos (and even images to a lesser extent ) on your computer would require a prohibitively large amount of resources were it not for vectors. You have linear algebra to thank (among many other contributors) every time you hear Sal's ever-patient disembodied voice.(7 votes)

- is the unit vector 'i' always equal to (1,0) or can it also have an arbitrary value?(3 votes)
- No, the unit vector i always has the value <1,0>. This is to indicate a direction relative to some axis system. Think of i as a way of saying East on a graph using vectors.(7 votes)

- I do understand this right now, but how can I stick it in my brain ?(2 votes)
- Four things that can help:

1. Get plenty of sleep before you learn or review (sleep has a strong relationship to ability to learn, think, or concentrate: https://youtu.be/09mb78oiPkA?t=42m18s (<-- this is really interesting).

2. Exercise for at least 10 minutes immediately before you learn or review (aerobic exercise immediately before mental activity has been shown to help with recalling memory and learning new things).

3. Practice the concept 1 day after first learning, then 3 days then a week later (Having gradually extending periods between using a concept helps to move the memory from short term to long term memory (possibly most important factor to make something stick).

4. Problem solve using the concept and if you can give yourself a dopamine reward immediately after successfully using the concept (Learning has been shown to be emphasised when the brain can make a clear correlation between an activity and a reward the body/brain wants).

Sorry for not having great links for each point, but this is what I have come across in my journey so far, and has worked well for me. Everything seems to be backed up with scientific evidence.(8 votes)

## Video transcript

We've already seen
that you can visually represent a vector as an arrow,
where the length of the arrow is the magnitude of the vector
and the direction of the arrow is the direction of the vector. And if we want to represent
this mathematically, we could just think
about, well, starting from the tail of
the vector, how far away is the head of the vector
in the horizontal direction? And how far away is it in
the vertical direction? So for example, in the
horizontal direction, you would have to
go this distance. And then in the
vertical direction, you would have to
go this distance. Let me do that in
a different color. You would have to go this
distance right over here. And so let's just say
that this distance is 2 and that this distance is 3. We could represent
this vector-- and let's call this vector v. We
could represent vector v as an ordered list or a
2-tuple of-- so we could say we move 2 in the
horizontal direction and 3 in the vertical direction. So you could represent
it like that. You could represent
vector v like this, where it is 2
comma 3, like that. And what I now want
to introduce you to-- and we could come up with
other ways of representing this 2-tuple-- is
another notation. And this really
comes out of the idea of what it means to
add and scale vectors. And to do that,
we're going to define what we call unit vectors. And if we're in
two dimensions, we define a unit vector for
each of the dimensions we're operating in. If we're in three dimensions,
we would define a unit vector for each of the three dimensions
that we're operating in. And so let's do that. So let's define a unit vector i. And the way that we denote
that is the unit vector is, instead of putting
an arrow on top, we put this hat on top of it. So the unit vector
i, if we wanted to write it in this
notation right over here, we would say it only goes 1 unit
in the horizontal direction, and it doesn't go at all
in the vertical direction. So it would look
something like this. That is the unit vector i. And then we can define
another unit vector. And let's call
that unit vector-- or it's typically
called j, which would go only in the
vertical direction and not in the horizontal direction. And not in the
horizontal direction, and it goes 1 unit in
the vertical direction. So this went 1 unit
in the horizontal. And now j is going to go
1 unit in the vertical. So j-- just like that. Now any vector, any
two dimensional vector, we can now represent as a sum of
scaled up versions of i and j. And you say, well,
how do we do that? Well, you could imagine
vector v right here is the sum of a vector
that moves purely in the horizontal direction
that has a length 2, and a vector that moves purely
in the vertical direction that has length 3. So we could say
that vector v-- let me do it in that
same blue color-- is equal to-- so if we want
a vector that has length 2 and it moves purely in
the horizontal direction, well, we could just scale
up the unit vector i. We could just
multiply 2 times i. So let's do that-- is equal
to 2 times our unit vector i. So 2i is going to
be this whole thing right over here or
this whole vector. Let me do it in
this yellow color. This vector right over
here, you could view as 2i. And then to that, we're going to
add 3 times j-- so plus 3 times j. Let me write it like this. Let me get that color. Once again, 3 times
j is going to be this vector right over here. And if you add this yellow
vector right over here to the magenta vector,
you're going to get-- notice, we're putting the tail
of the magenta vector at the head of
the yellow vector. And if you start at the
tail of the yellow vector and you go all the way to the
head of the magenta vector, you have now constructed
vector v. So vector v, you could represent it as a
column vector like this, 2 3. You could represent
it as 2 comma 3, or you could represent it as
2 times i with this little hat over it, plus 3 times j,
with this little hat over it. i is the unit vector in
the horizontal direction, in the positive
horizontal direction. If you want to go
the other way, you would multiply it by a negative. And j is the unit vector
in the vertical direction. As we'll see in future
videos, once you go to three dimensions,
you'll introduce a k. But it's very natural to
translate between these two things. Notice, 2, 3-- 2, 3. And so with that, let's actually
do some vector operations using this notation. So let's say that I
define another vector. Let's say it is vector b. I'll just come up with
some numbers here. Vector b is equal to negative 1
times i-- times the unit vector i-- plus 4 times the unit vector
in the horizontal direction. So given these two
vector definitions, what would the would be the
vector v plus b be equal to? And I encourage you to pause
the video and think about it. Well once again,
we just literally have to add
corresponding components. We could say, OK,
well let's think about what we're doing in
the horizontal direction. We're going 2 in the
horizontal direction here, and now we're going negative 1. So our horizontal
component is going to be 2 plus negative
1-- 2 plus negative 1 in the horizontal direction. And we're going to multiply
that times the unit vector i. And this, once
again, just goes back to adding the corresponding
components of the vector. And then we're going to have
plus 4, or plus 3 plus 4-- And let me write it that way--
times the unit vector j in the vertical direction. And so that's going
to give us-- I'll do this all in this one color--
2 plus negative 1 is 1i. And we could literally
write that just as i. Actually, let's do that. Let's just write that as i. But we got that from 2
plus negative 1 is 1. 1 times the vector is just
going to be that vector, plus 3 plus 4 is 7-- 7j. And you see, this is exactly
how we saw vector addition in the past, is that we
could also represent vector b like this. We could represent it
like this-- negative 1, 4. And so if you were
to add v to b, you add the corresponding terms. So if we were to add
corresponding terms, looking at them as column
vectors, that is going to be equal to 2 plus
negative 1, which is 1. 3 plus 4 is 7. So this is the exact same
representation as this. This is using unit
vector notation, and this is representing
it as a column vector.