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Unit Vectors - Concept
One special type of vectors are unit vectors. Unit vectors have a magnitude of one, and the specific unit vectors i and j represent the positive horizontal and vertical unit vectors in a two dimensional plane. We can write any two dimensional vector in terms of these two unit vectors, and that notation is seen often in Calculus.
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.