Unit vectors

we covered the idea of vector length vector length many many videos ago and I realized that I forgot to cover an important topic and this topic is going to be useful when we do some types of transformations actually the projections that I'll do in the next video and the notion that I forgot to do is the notion of a unit vector unit vector and all this is is a vector this is just a vector that has a length of 1 so let me see has length and we've defined length it has a length of 1 so if something is a unit vector let's say that you right here is a unit vector and it can and it's a member of RN it's a member of RN then that means that if we have u u looks like this has n components u 2 all the way to u n we know what the length of this is right the length of this the definition of the length we know that the length of U sometimes called the norm of U it's just equal to the square root of the squared sums of all of its components and if you think about it this is just an extension of the Pythagorean theorem to some degree but so it's u1 squared plus u2 squared all the way to u n squared and it's the square root of that if this is a unit vector if this is a unit vector so this is a unit vector unit vector that implies that the length of U will be equal to 1 and that doesn't matter in what dimension space we are this could be our 100 this could be our two for it to have a unit vector in any of those pieces spaces their length is one so the next obvious question is how do you construct a unit vector so let's say that I have some vector V and let's say it's not a unit vector so it's v1 v2 all the way to VN and I want to turn it into some vector u that is a unit vector that just goes in the same direction so you we'll go in the same direction same direction as V but just has a length the length of U is going to be equal to 1 how do i construct this vector u here well what I could do is I could take the length of V I could find out what the length of V is and we know how to do that we just apply this definition of vector length and what happens if I figure out the length of V and then I multiply the vector V times that so what if I make my U what if I say U is equal to 1 over the length of V 1 over the length of V times V itself what happens here if I take the length of this thing right here if I take the length of that what do I get so this is so the length of U is equal to the length of this scalar remember this is just some number right it's equal to this scalar and I'm assuming V is a nonzero vector so the length of this lowers whatever this scalar number is times V and we know that we can take this scalar out of the formula we could show that I think I've shown it in a previous video that the length of C times V is equal to C times the length of V let me write that down and that's essentially what I'm assuming right here that if I take the length of C times some vector V that is equal to C times the length of V I think we showed this when we first were introduced to the idea of length so we know that this is going to be equal to 1 over the length of vector V that's my C times so this thing right here is that thing right there times this thing times the length of vector V well what's this going to be equal to 1 over something times that's something well this is just going to be equal to 1 so that's all a unit vector is if you want to find a unit vector or sometimes it's called a normalized vector it's nice to another normalized normalized vector that goes in the same direction as some vector V you just figure out the length of V just using the definition of vector length in RN and then multiply one over that length times the vector V and this is just a scalar and then you get your vector you let me do an example just to make sure you get the idea so let's say I have some vector let's say I have some vector V let's say I have some vector V and it's in r3 let's say it's 1 2 minus 1 what is the length of V the length of V is equal to the square root of 1 squared plus 2 squared plus minus 1 squared minus 1 squared and that is equal to the square root of what 1 plus 1 plus 4 square root of 6 so that is the length of V so if I want to construct a normalized vector U that goes in the same direction as V I just take I can just define my vector U as being equal to 1 over the length of V 1 over the square root of 6 times V so times 1 2 minus 1 which is equal to 1 over square root of 6 2 over the square root of 6 and minus 1 over the square root of 6 and I'll leave it for you to verify that the length of U is going to be equal to 1 and I'll just draw it one other idea here that you'll often see when something is a unit vector instead of using this little arrow on top of the vector they'll often write a unit vector with a little hat on top of it like that that signifies that we're dealing with a unit vector and for those of you who've taken your vector calculus or I've done a little bit of engineering you're probably familiar with the vectors I J and K and the reason why they have this little hat here is because these are all unit vectors in r3 there are n members of our three and they're all unit vectors these are actually the basis vectors in r3 for those of you all have been watching my transformation videos these are equivalent to the vectors e1 which I could write with a hat on it really eetu and e3 which are the standard basis vectors in r3 anyway now that you've been exposed to it now I can start to use the idea of a unit vector in future videos