Unit Vectors

So one really important topic in units is the idea of a unit vector and all a unit vector is, is a vector with length 1. Now there are some special notation for unit vectors if you have a unit vector u instead of writing this half arrow on top of the u you'll often see it expressed as u hat that's how this is written u hat the little carat on top of it. So here are our 2 special unit vectors I hat and j hat, I hat has components 1, 0 and j hat has components 0, 1. Now any vector can be written in terms of i and j for example the vector 4 negative 3 can be written as 4, 0 plus 0 negative 3 and this is exactly 4 times i 4i hat plus and this is negative 3 times j and I can actually write minus 3j so this is how you would write this vector 4 negative 3 as a linear combination of the unit vectors i and j this is often the way vectors are expressed in Math classes, Science, Engineering. So let's do that for a bunch of vectors, let's express in terms of the unit vectors i and j.

Now you'll be able to see how these are expressed in terms of i and j pretty quickly but just to remind you, you break it up so that you've got the horizontal component by itself and 0 and then 0 and the vertical component and then this is exactly 6 times i hat plus and this is exactly 13 times j hat. And then here I won't bother to separate you can probably tell if this is negative 9 times i hat plus 0 times j hat which is just negative 9 i hat it's not necessary to write the plus 0.

Now here you'll want to multiply this out and simplify before you convert it to i and j form. So let me distribute this scalar of multiple 2 times 8 gives me 16, 2 times negative 11 negative 22 minus 3 times 5 is 15 and 3 times negative 8 is negative 24. So subtracting I get 16 minus 15 are 1, negative 22 plus 24 so that's 2 and this becomes i 1 times i is i plus 2 j alright that's it. These are 2 different but very closely related forms for vectors you can write them in component form or linear combinations of the unit vectors i and j.