Unit vectors and engineering notation

what I want to do in this video is show you a way to represent a vector by its components and this is sometimes called engineering notation for vectors but it's 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 let's break down this vector right over here I'm just assuming it's a velocity vector vector V it's magnitude is 10 meters per second and it's pointed in a direction 30 degrees above above the horizontal so we've 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 sine of 30 degrees is going to be 10 meters per second times the sine of 30 degrees sine of 30 degrees this comes from the basic trigonometry from sohcahtoa and I cover that in more detail in previous videos sine of 30 degrees is one-half so this is going to be five or five meters per second ten times one half is five five meters per second so that's 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 use this notation which isn't as tangible as I like and that's why I'm gonna make it a little bit better in this video I said that that vector itself is five meters per second five meters per second but what I told you is 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 it's positive it means up and if it's negative it means down so I kind of had to give you this context here so that you could appreciate that this is a vector that just the sine of it is giving you its direction but I had to keep telling you this is a vertical vector so it was a little bit it wasn't that tangible and so we had the same issue when we talked about the way the same issue we talked about the horizontal vectors so this horizontal vector right over here the magnitude of it the magnitude of this horizontal vector is going to be ten cosine of 30 degrees and once again it comes straight out of basic trigonometry and cosine of 30 degrees and so cosine of 30 degrees is square root of 3 over 2 square root of 3 over 2 multiply it by 10 you get 5 square roots of 3 5 square roots of 3 meters per second and once again in previous videos I said look this is actually I used this notation some times where I was actually saying that the vector is 5 square roots of 3 meters per second but in order to ensure that this wasn't just a magnitude I kept having to tell you in the horizontal direction if it's positive it's going to the right and if it's negative it's going to the left 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 it 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'll introduce the vector I the vector I sometimes it's called I hat and I'll draw it like here so the vector I'll make it a little bit smaller so the vector I hat so that right there is a picture of the vector I hat and we put that little hat on top of the eye to show that it is a unit vector and what a unit vector is so I hat goes in the positive x-direction that's just how it's defined and we also the unit vector tells us that it's magnitude is 1 so I the magnitude of the vector I hat is equal to 1 and its direction is in the positive x-direction so if we really wanted to specify this the this the X this this kind of X component vector in a better way we really should call it we really should call it 5 square roots of 3 times this unit vector because it's 5 this green vector over here is going to be 5 square roots of 3 times this vector right over here because this vector just has length 1 so it's 5 square roots of 3 times the unit vector and what I like about this is now I don't have to tell you remember this is a horizontal vector or positive is positive is to the right negative to the left it's implicit here because clearly if this is a positive value it's going to be a it's going to be a positive multiple of I it's going to go to the right if it's an negative value it flips around the vector and then it goes to the left so this is actually a better way of specifying of specifying the x-component vector or if I broke it down this vector V into its X component 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 well 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 one this little hat on top of it tells us or sometimes it's called a character a carrot character that tells us that it is a vector but it is a unit vector it has a magnitude of one and by definition the vector J goes in is is has a magnitude of 1 in the positive y direction so this the Y component of this vector instead of saying it's 5 meters per second in the upward direction or instead of saying that it's implicitly upwards because it's a vertical vector or it's a vertical component and it's positive we can now be a little bit more a little bit more specific about it we could say that it is equal to it is equal to 5 times J 5 times J because you see this magenta vector it's going the exact same direction as J the exact same direction is J it's just 5 times longer I don't know if it's exactly 5 times I'm trying to estimate it right now it's just 5 times longer now what's really cool about this is besides just being able to express the components as now multiples of explicit vectors instead of just being able to do that which we did do we're 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 vector 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 won't we don't always have to draw it like this so we can write that vector V is equal to it's equal to vector we write it this way it's equal to its x-component vector plus the y-component vector plus the y-component vector and we can write that the x component vector is 5 square roots of 3 5 square roots of 3 times i 5 square roots of 3 times i and then it's going to be plus the plus the y component the vertical component which is 5 j which is 5 5 times j and so what's really neat here is now you can specify any vector in two dimensions by some combination of eyes and JS or some scaled up combinations of eyes and js and if you want to go into three dimensions if you want to go to three dimensions and you often will as especially as the physics class moves on through the year you can introduce a vector in you can introduce a vector in the positive z direction depending on how you want to do it although z is normally up-and-down but whatever the next dimension is you can define a define a vector K that goes into that third dimension here I'll do it in a kind of unconventional way I'll make K go in that direction although the standard convention when you do three dimensions is that K is the up-and-down dimension but this by itself is already pretty neat because we can now represent any vector any vector through its components and it's also going to make the math much easier