Two-dimensional projectile motion
-
Visualizing Vectors in 2 Dimensions
-
Projectile at an Angle
-
Different Way to Determine Time in Air
-
Launching and Landing on Different Elevations
-
Total Displacement for Projectile
-
Total Final Velocity for Projectile
-
Correction to Total Final Velocity for Projectile
-
Projectile on an Incline
-
Unit Vectors and Engineering Notation
-
Clearing the Green Monster at Fenway
-
Green Monster at Fenway Part 2
-
Unit Vector Notation
-
Unit Vector Notation (part 2)
-
Projectile Motion with Ordered Set Notation
Unit Vectors and Engineering Notation Using unit vectors to represent the components of a vector
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- What I want to do in this
- video is show you a way to
- represent a vector by its component
- And this is sometimes called engineering notation for vectors.
- But its super useful because
- it allows us to keep track
- of the components of the vector, and
- it makes it a little bit tangible when we
- talk about the individual components.
- So lets break down this vector right over here.
- I'm just assuming it is a velocity vector, vector v ,
- its magnitude is 10 m/s and its pointed in the direction
- 30 degrees above, above the horizontal.
- So we have broken down these
- vectors in the past before.
- The vertical component right
- over here, its magnitude,
- its magnitude would be, so, the magnitude of
- the vertical component right over here,
- is going to be 10 sin of 30 degrees,
- is going to be 10 meters per sec times
- the sin of 30 degrees
- , sin of 30 degrees, this just comes
- from basic trigonometry from soh cah toh,
- and I covered that in more details
- in previous videos
- sin of 30 degrees is 1/2
- So this is going to be 5 or 5 meters per second
- Ten times 1/2 is 5, 5 meters per second so
- that is the magnitude of its vertical component
- And in the last few videos I kind of
- , in a less tangible way of specifying
- the vertical vector, I often used this notation
- which isn't that tangible as I like it,
- that's why I am going to make it little bit
- better in this video.
- I said that the vector
- its self is 5 meters per sec, 5 meters per sec
- but what I told you that the direction is
- implicitly given because this
- is a vertical ,
- this is a vertical vector and I told you
- in previous videos that
- if its positive, it means up
- and if its negative its means down.
- So I kind of have to give you this context
- here so that you could appreciate
- that this is a vector, that just the sign
- of it is giving you its direction
- But I have to keep telling you this
- a vertical vector.So its a little bit
- it wasn't that tangible,and so we had the same issue,
- when we talked about the
- we had the same issue talked about
- the horizontal vectors, so this horizontal vector
- right over here, the magnittude of it,
- the magnitude of this horizontal vector is going
- to be 10 cosine of 30 degres.
- And once again comes straight out of basic trigonometry.
- tan cosine of 30 degrees and so
- cosine of 30 degres is
- square-root of three over 2
- square root of 3 over 2.
- multiply it by ten, you get
- 5 square roots of 3 meters per sec.
- And once again in previous videos
- I said, look this is actually
- I used this notation sometimes
- where I was actually saying the vector is
- 5 square root of 3 metres per sec
- but in order to ensure that this wasn't not just the
- magnitude I kept having to tell you that
- in the horizontal direction if its positive
- , its going to the right and
- if its negative its going to the left.
- But what I want to do in this video
- is give us a convention so that I don’t have to
- keep doing this for the direction
- and it all, it makes all a little bit more tangible
- And so what we do is we introduce
- the ideas of, or the idea of unit vectors.
- Of unit vectors.
- so by definition we introduce the
- vector i, the vector i, sometimes its called
- i hat, and I'll draw it like here.
- So the vector.Let me make it a little bit smaller,
- So the vector i hat,
- so that right there is a picture of the vector i hat
- we put a hat on top of i
- to show that it’s a unit vector.
- And what a unit vector is,
- so the i hat vector goes in the
- positive x -direction.
- That‘s just how its defined
- and we also, unit vector tells us
- that its magnitude is one.
- So, the magnitude of the vector i hat
- is equal to one and its direction
- is in the positive x -direction.
- So if we really wanted to specify
- this kind of x -component vector in a better way.
- We really should call it
- , we really should call it,
- five square roots of 3 times this unit vector.
- Because it 5, this green vector over here
- is going to be 5 squared roots of 3
- times this vector right over here.
- cause' this vector just has length 1.
- So its 5 squared of 3 times the unit vector.
- and what I like about this is that
- now I don’t have to, tell you
- Remember this a horizontal vector,
- positive is,
- positive is to the right and
- negative to the left,
- It’s implicit here,
- because clearly if it’s a positive value
- Its going to be a positive multiple of i,
- its going to go to the right
- If its a negative value
- it flips around the vector and
- its goes to the left .
- So this is a actually a better way of specifying,
- of specifying,
- the x component vector
- or if I broke it down this vector v,
- into its x components
- this is a better way of specifying that vector.
- Same thing for the y -direction,
- We can define a unit vector
- and let me pick a color,
- that I have not used yet,
- let me find a, oh, this pink I haven’t used.
- We can define a unit vector
- that goes straight up in the
- y-direction called unit vector j
- and once again the magnitude of unit vector j
- is equal to 1
- This little hat on top of it tells us
- or sometimes is called a caret,
- a caret character,
- tells us that it is a vector but
- it is a unit vector
- and has magnitude of 1.
- And by definition the vector j
- goes in, has a magnitude of 1
- in positive y-direction, so this
- the y -component of this vector,
- instead of saying its,
- 5 meters per second in upwards direction
- and instead of saying that its implicitly upwards
- because the vertical vector or its
- vertical component in its positive,we can now be a little bit more
- Or a little bit more specific about it ,we could
- say it’s a equal to
- equal to 5 times j
- , 5 times j
- because you see this magenta vector,
- is going the exact same direction as j
- , the exact same direction as j
- it is just 5 times longer,
- I don’t know if its exactly 5 times,
- I'm trying to estimate it right now.
- Its 5 times longer
- Now what's really cool about this, is besides
- just being able to express the components as
- now multiple of explicit vectors,
- instead of just being able to do that
- which we did do, or we are
- representing the components as explicit vectors
- we also know that the vector, v
- is the sum of its components,
- if you add, if you start with this, this
- green vector right here
- and you add this vertical component
- right over here you have head to tails
- you get, you get the blue vector,
- and so we can actually use the components
- to represent the vector itself
- we don't always have to draw like this
- So we can write,
- that vector, v is equal to
- its equal to vector,
- let me write it this way, is equal to its x-component
- vector plus the y- component vector
- , plus the y- component vector,
- And we can write that, x-component vector
- is 5 square roots of 3 times i
- , 5 square roots of 3 times i,
- and then its going to be plus
- the y component, the vertical component
- which s five times j,
- which s five times j
- and so what's really neat here
- is now you can specify any vector
- in two dimensions,
- by some combintion of i and j
- scaled up combination of i and j
- and if you want to go in three dimension, and you
- often will,
- as specially physics class moves on through the year
- you can introduce a vector in the positive z-direction
- depending upon how you want to do it,
- although z is normally up and down,
- but whatever the next dimension is
- you can divine, divide a vector k
- that goes into that third dimension
- here I will do it in a kind of unconventional way
- I'll make k go in that direction.
- Although the standard convention when you do
- in the three dimensions is that k is the
- up and down dimension.
- But this by itself is already petty neat because
- we can now represnt any vector,
- any vector through its components
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.