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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.

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  • blobby green style avatar for user LedimirN
    Don't you have to square 5*sqrt3 and 5 to get the resultant vector v?
    (14 votes)
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    • mr pants teal style avatar for user erikgaas
      You are thinking a^2 + b^2 = c^2, and you would be correct in thinking about it, but remember that c^2 is not the length of the hypotenuse. c is the length of the hypotenuse. However, because we can use unit vectors and trig identities we can merely equate the hypotenuse to the sum of the i and j vectors. If you take the length of 5*sqrt(3) and 5 as sides of a right triangle by plugging those into the Pythagorean theorem you should find that c^2 = 100, so c equals 10, the value we originally worked with. So no, it would be wrong to square the vectors at the end of the problem. Sal's trig playlist should make things more concrete.
      (20 votes)
  • blobby green style avatar for user Gabriel
    So the unit vector notation is the same as engineering notation for vectors?
    (10 votes)
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    • leaf green style avatar for user kadeemwilliams89
      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.
      (7 votes)
  • blobby green style avatar for user Rahul Chelseafc Abhishek
    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)
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  • hopper cool style avatar for user ★✮✶Arsh Pervez✶✮★
    If you added a i vector and a j vector what would it equal?
    (3 votes)
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  • blobby green style avatar for user upendrakepler
    How is I and j vector are practically used in our life (like spacecraft moving or launched with help of vector calculations)
    (6 votes)
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  • old spice man green style avatar for user M.Sanjay Rangharajhan
    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)
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    • orange juice squid orange style avatar for user Brandon Baker
      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)
  • male robot donald style avatar for user Dwij Hirpara
    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)
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    • aqualine ultimate style avatar for user Kip Rodes
      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)
  • blobby green style avatar for user shaakirahlee
    So how do you know weather to use sine or cosine when dealing with Vx and Vy??
    (2 votes)
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  • leaf green style avatar for user arjun shankar
    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)
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  • blobby green style avatar for user amanda luna
    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)
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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.