Unit vector notation (part 2)

welcome back in the last video I just you know at the end of the video like I always do an attempt to confuse you I told you that if I had two vectors and let me just make up some new ones just so I can draw them visually you know in a second or two let's call the first vector a let me do a different color this toothpaste color is getting monotonous let me do something that looks relaxing so let's call a first vector a and I don't know let's say it's let me put some make it interesting lets me say it's -3 times the unit vector I plus 2 times the unit vector J and then I have vector B and that is equal to I don't know let's say it's 2 I 2 times the unit vector I plus I don't know let me say 4 times the unit vector J in the last video I said well the whole reason why this unit vector notation is even well one of the reasons what we'll see that there are many reasons why it's useful but one of the really cool things about it is before when we added vectors we would put them head to tails and then draw it visually and then we'd have this new vector and we really had no way of expressing it without drawing it but when we write things as multiples of the unit vectors we don't have to draw it and it's actually very easy to add vectors and how do we do it we just add the X components and we add the Y components so we said that these these two vectors a plus B these little weird arrows on top that's just saying that those are vectors did that equals oh this should be an equal sign equals not not equals that's equals so it's minus 3 plus 2i and I'm going to arbitrary switch colors because it's getting monotonous plus 2 plus 4j we just added the X components or the multiples of I and we added the Y components or just the multiples of J right because I was the unit vector in the X direction and J was unit vector in the Y direction and we get what's minus 3 plus 2 that's minus 1 so we get minus 1 I or that could just be minus I but I'll write the 1 just because we're just getting warmed up with unit vector so minus 1 I +6 J and when I did that you might say well Sal you know I can't just take your word for it because you you've seen kind of a you know not not someone who should should be believed blindly so and I think that's a valid opinion to have so I will show you that this works by doing by adding the vectors visually so let's let's draw it and I think this will give you a little better sense of of unit vectors generally let me draw the axes the axes so that's my X that's my y-axis and let me draw my x-axis to make sure I have enough space to draw the unit vectors that we've drawn under draw the vectors that we've drawn so that's just to show you that the axes go on forever I have to draw that arrow alright so let's see I go so let's say this is 1 2 3 this is 1 2 3 4 I draw 1 2 3 4 5 6 I think we should be able to now add them I didn't have to waste all this space down here so let's just first draw the vectors minus 3i plus 2j so minus 3i so minus 3i just this this right here is going to be a vector that looks something like this so it'll go it's just minus 3 times the X vector so it'll go to the left right because I is the pause is 1 in the positive direction so if we if we put a negative there it flips it over let me do it in a different color so this is minus 3i and then plus 2j so plus 2j looks like this alright this is plus 2j and if we were add those two vectors visually we can put them head to tails and the way we could do that we could either shift this vector up like this and draw it up here we could shift this vector and put its tail to this vectors head but either way so let's shift the one up so if we shift it up like that now remember we're just doing the head to tails visual addition method of vectors so I just put this tail to this head and what do we get so vector a will look like this and I'm going to do it in the same color as vector a just so because I have a feeling that this diagram might get complicated well if I wanted to use the line tool it undo into the line tool okay so this is vector a that's what vector a it looks like and so we worked backwards I gave you the X component and the y component and then I added them together by doing the head to tails method and so this is what vector a would look like and instead of drawing it a very easy representation is exactly what we did up here the unit vector notation and what's a vector B look like so it's two I'm going to do a completely different color it's 2i so it's this vector two times the unit vector I that's this plus four J 4 J 1 2 3 4 so it looks like this and if let's take this one and shift it over to the left so we can put its tail to this vectors head so would we look like this so vector B will look and I'll do it in red and I'll use the line tool vector B looks like this all right I just put its components head to tails and that's how I got vector B and if I were to add them visually I would do it the same way that I added its components I would put the tail of one vector to the head of the other and see get the resulting vector so we can do it either way let's shift this a vector let's shift it in this direction and put it because remember vectors we're just giving a magnitude in a direction we're not telling you we're not necessarily giving a starting point so you can shift them you just can't change their orientation or the magnitude and that's actually how you add them you've shift them and put them head to tails that's when you add them visually so let's put that a vector let's put it up here so if we have the a vector it looks something like this and I know my the a vector we'll look something like this and I want it to work out right some so the a vector something like that and remember all I did is I took the same vector and I just shifted it so that it can start at the head so its tail can start at the head of the B vector right so I just shifted the a vector so this is still the a vector by moving a vector around you haven't changed the vector I would only change the vector if I scaled it if I made it bigger or smaller if I if I change this to orientation and so visually this is B this is a so if I add a to be the resulting vector going head to tails I'll do it in in let me see elders in the screen color would look like this it would look like that right so here we took all this trouble and I had to draw these straight lines to visually add these two vectors this green vector is a plus B let's see if this green vector is the same thing that we got that we got here let's see if it's the same thing as this so we got negative one times I so negative one is like here and then we have six j6j let me do it in another color 6j would look like this 6j looks like that you put them heads to tails and it would get would be something like this and that is the green vector and actually just so you know I know it didn't line up perfectly and that's because I'm not drawing neatly but these two points should actually be here if I were to have drawn this better but I know this was very confusing I had all these colors but the whole point of it is I wanted you to show that you know you could visually draw vectors and then have you know shift them around and then put them heads to tails and then get the resulting vector that's one way to add vectors and there's still real no way to end the analytically represented it or you could just write any vector as it's x and y-components and then the sum of the vectors is just going to be the sum of the x's and then the sum of the Y's and that's a much cleaner and a much easier and much less prone to error way of adding or subtracting really two vectors so hopefully that was convincing that this that a plus G a plus B really is this vector if it wasn't I'm sorry and I hope I didn't confuse you more but now that we have this out of the way and hopefully you're convinced that unit vector notation is useful we can move on and maybe try to do some of our old projectile motion problems using this notation and maybe it'll let us to do a little bit extra stuff with it see you soon