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### Course: Physics library > Unit 2

Lesson 1: Two-dimensional projectile motion- Horizontally launched projectile
- What is 2D projectile motion?
- Visualizing vectors in 2 dimensions
- Projectile at an angle
- Launching and landing on different elevations
- Total displacement for projectile
- 2D projectile motion: Identifying graphs for projectiles
- 2D projectile motion: Vectors and comparing multiple trajectories
- What are velocity components?
- Unit vectors and engineering notation
- Unit vector notation
- Unit vector notation (part 2)

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# Unit vector notation

Learn how to factor in magnitude and direction when adding and subtracting vectors. See how to break vectors into x and y components, and how to use unit vector notation to label vectors in a way that represents them more efficiently and analytically, making it easier to add and subtract them. Created by Sal Khan.

## Want to join the conversation?

- So the unit vector notation is the same as engineering notation for vectors?(11 votes)
- Yes,but this similarity is in their conceptualizations:

-Engineering Notation is the representation of a ''vector'' by its individual components.

-And as such by definition Unit vector notation is the analytically representation of 2 dimensional vector - in that, any 2-D vector can be represented by any combination of these U.Vectors

One could then say that they fall in the same category.(8 votes)

- what's the answer to this question ?

-> a = i^ + j^ - K^, b = i^ - j^- k^

Find = a-b

P.S.- can't properly write i cap, j cap, k cap. I hope you understand!(0 votes) - If you added a i vector and a j vector what would it equal?(3 votes)
- It will be i + j(7 votes)

- How is I and j vector are practically used in our life (like spacecraft moving or launched with help of vector calculations)(6 votes)
- So V=5sqrt3i + 5j. But in the question it is given that V=10. And, 5sqrt3 + 5 is not equal to 10. Can somebody please explain?(4 votes)
- I think what you're missing (it isn't well explained) is that unit vector notation is an alternate way to define a vector, not an actual equation. The i-hat, j-hat, k-hat etc don't just disappear and allow you to add the values to get the magnitude of v. You need to forget about the plus or minus sign as an operator in this and think of it as an indicator of positive or negative values. To visualize what I'm saying, try using a comma in the unit vector notation: (+5) i hat, (-3) j hat. The plus and minus are not operators when you use a comma and parentheses like this, and that is where you need to bring your brain. It is a shorthand way of writing out the individual components of a vector, which becomes very useful when manipulating multiple vectors. Add a vector with a magnitude 10 at 30 degrees to a vector of magnitude 5 at 90 degrees. Since you already need to break the vectors into components to solve the problem, 5sqrt3 ihat+5 jhat, and 0 ihat+5jhat, you can present the vector as 5sqrt3 ihat+10 jhat. this is simpler to work with than the magnitude degree calculation of magnitude is sqrt(5sqrt3^2+10^) and direction is arctan 10/5sqrt3. Both define the same vector, the vector notation is simply cleaner and more easily worked with.(3 votes)

- So if sum two vectors we join them head to tail? and what if i wanna subtract them (visually/in a diagram or something)?(3 votes)
- To subtract or add them we use the tail to tip method where we put the head of one on the tail of another. The reason for this is that there are no negative vectors or lines, they just go in different directions. Simply, there is no subtraction or negatives.(4 votes)

- So how do you know weather to use sine or cosine when dealing with Vx and Vy??(2 votes)
- you need to draw your triangle and figure out whether the side you are trying to figure out is opposite or adjacent. If it's opposite, you are going to use sine. Adjacent will be cosine.(5 votes)

- Why does Sal tend to use an angle off of the x-axis when talking vectors as opposed to the more conventional y-axis?(2 votes)
- In physics the convention is to use the x-axis, probably because it's easier to measure the angle from the horizontal in real life (the x-axis would be the ground, or the top of a table, or whatever) than to measure the angle from the vertical (which would be straight upward and more difficult to find a line to measure against). I've seen the y-axis convention more in trig, usually in navigation-related applications in which the y-axis represents north.(4 votes)

- can someone clear my doubt?

imagine i have 3 points in xyz space.. and those points are collinear.......

if the vector positions of those 3 points have different direction.... would those vector's would be termed as collinear? since those points are collinear??(3 votes)- vectors will not be termed as collinear unless they pass from origin.and even from origin they may be termed collinear only when they are from opposite directions.

Hope this helps(1 vote)

- if i am given two vectors how would i find the magnitude and direction of the resultant using the rectangular components method. How do i solve this(3 votes)
- Then just add the rectangular components.then to find the magnitude, use the pythagoras theorem and for direction,use the slope of the tangent line.(2 votes)

## Video transcript

Good afternoon. We've done a lot of
work with vectors. In a lot of the problems, when
we launch something into--- In the projectile motion problems,
or when you were doing the incline plane
problems. I always gave you a vector, like I would draw
a vector like this. I would say something
has a velocity of 10 meters per second. It's at a 30 degree angle. And then I would break it up
into the x and y components. So if I called this vector v, I
would use a notation, v sub x, and the v sub x would have
been this vector right here. v sub x would've been this
vector down here. The x component of the vector. And then v sub y would have been
the y component of the vector, and it would have
been this vector. So this was v sub x,
this was v sub y. And hopefully by now, it's
second nature of how we would figure these things out. v sub
x would be 10 times cosine of this angle. 10 cosine of 30 degrees, which I
think is square root of 3/2, but we're not worried about
that right now. And v sub y would be 10 times
the sine of that angle. This hopefully should be
second nature to you. If it's not, you can just go
through SOH-CAH-TOA and say, well, the sine of 30 degrees
is the opposite of the hypotenuse. And you would get
back to this. But we've reviewed all of that,
and you should review the initial vector videos. But what I want you to do now,
because this is useful for simple projectile motion
problems-- But once we start dealing with more complicated
vectors-- and maybe we're dealing with multi-dimensional
of vectors, three-dimensional vectors, or we start doing
linear algebra, where we do end dimensional factors --we
need a coherent way, an analytical way, instead of
having to always draw a picture of representing
vectors. So what we do is, we use
something I call, and I think everyone calls it, unit
vector notation. So what does that mean? So we define these
unit vectors. Let me draw some axes. And it's important to keep in
mind, this might seem a little confusing at first, but this
is no different than what we've been doing in our physics
problem so far. Let me draw the axes
right there. Let's say that this is 1,
this is 0, this is 2. 0, 1, 2. I don't know if must been
writing an Arabic or something, going backwards. This is 0, 1, 2,
that's not 20. And then let's say this is 1,
this is 2, in the y direction. I'm going to define what I call
the unit vectors in two dimensions. So I'm going to first
define a vector. I'll call this vector i. And this is the vector. It just goes straight in the x
direction, has no y component, and it has the magnitude of 1. And so this is i. We denote the unit vector
by putting this little cap on top of it. There's multiple notations. Sometimes in the book, you'll
see this i without the cap, and it's just boldface. There's some other notations. But if you see i, and not in
the imaginary number sense, you should realize that that's
the unit vector. It has magnitude 1 and it's
completely in the x direction. And I'm going to define another
vector, and that one is called j. And that is the same thing
but in the y direction. That is the vector j. You put a little cap over it. So why did I do this? Well, if I'm dealing with
two dimensions. And as later we'll see in three
dimensions, so there will actually be a third
dimension and we'll call that k, but don't worry about
that right now. But if we're dealing in two
dimensions, we can define any vector in terms of some sum
of these two vectors. So how does that work? Well, this vector here,
let's call it v. This vector, v, is
the sum of its x component plus its y component. When you add vectors, you
can put them head to tail like this. And that's the sum. So hopefully knowing what we
already know, we knew that the vector, v, is equal to its x component plus its y component. When you add vectors, you
essentially just put them head to tails. And then the resulting sum
is where you end up. It would be if you added this
vector, and then you put this tail to this head. And you end up there. So you end up there. So that's the vector. So can we define v sub x as some
multiple of i, of this unit vector? Well, sure. v sub x completely goes
in the x direction. But it doesn't have
a magnitude of 1. It has a magnitude of 10
cosine 30 degrees. So its magnitude is ten. Let me draw the unit
vector up here. This is the unit vector i. It's going to look something
like this and this. So v sub x is in the exact same
direction, and it's just a scaled version of
this unit vector. And what multiple is it
of that unit vector? Well, the unit vector has
a magnitude of 1. This has a magnitude of 10
cosine of 30 degrees. I think that's like, 5
square roots of 3, or something like that. So we can write v sub x-- I keep
switching colors to keep things interesting. We can write v sub x is equal
to 10 cosine of 30 degrees times-- that's the degrees
--times the unit vector i-- let me stay in that color, so
you don't confused --times the unit vector i. Does that make sense? Well, the unit vector i goes in
the exact same direction. But the x component of this
vector is just a lot longer. It's 10 cosine 30
degrees long. And that's equal to-- cosine of
30 degrees is square root of 3/2 --so that's 5 square
roots of 3 i. Similary, we can write the y
component of this vector as some multiple of j. So we could say v sub y, the y
component-- Well, what is sine of 30 degrees? Sine of 30 degrees is 1/2. 1/2 times 10, so this is 5. So the y component goes
completely in the y direction. So it's just going to be a
multiple of this vector j, of the unit vector j. And what multiple is it? Well, it has length 5,
while the unit vector has just length 1. So it's just 5 times
the unit vector j. So how can we write vector v? Well, we know the vector v is
the sum of its x component and its y component. And we also know, so this
is a whole vector v. What's its x component? Its x component can be
written as a multiple of the x unit vector. That's that right there. So you can write it as
5 square roots of 3 i plus its y component. So what's its y component? Well, its y component is just
a multiple of the y unit vector, which is called
j, with the little funny hat on top. And that's just this. It's 5 times j. So what we've done now, by
defining these unit vectors-- And I can switch this
color just so you remember this is i. This unit vector is this. Using unit vectors in two
dimensions, and we can eventually do them in multiple
dimensions, we can analytically express any
two dimensional vector. Instead of having to always draw
it like we did before, and having to break out
its components and always do it visually. We can stay in analytical mode
and non graphical mode. And what makes this very useful
is that if I can write a vector in this format, I can
add them and subtract them without having to resort
to visual means. And what do I mean by that? So if I had to find some vector
a, is equal to, I don't know, 2i plus 3j. And I have some other
vector b. This little arrow just
means it's a vector. Sometimes you'll see it
as a whole arrow. As, I don't know, 10i plus 2j. If I were to say what's
the sum of these two vectors a plus b? Before we had this unit vector
notation, we would have to draw them, and put them
heads to tails. And you had to do it visually,
and it would take you a lot of time. But once you have it broken up
into the x and y components, you can just separately add
the x and y components. So vector a plus vector b,
that's just 2 plus 10 times i plus 3 plus 2 times j. And that's equal
to 12i plus 5j. And something you might want to
do, maybe I'll do it in the future video, is actually draw
out these two vectors and add them visually. And you'll see that you
get this exact answer. And as we go into further
videos, or future videos, you'll see how this is super
useful once we start doing more complicated physics
problems, or once we start doing physics with calculus. Anyway, I'm about to run out
of time on the ten minutes. So I'll see you in
the next video.