in this video we continue with our study of vectors by particularly looking at their components and how we could use those components to compute magnitudes this is going to be used as an alternative to the graphical techniques we want some kind of formula based or symbol-based ways to manipulate these vectors as well one way to define vectors is using building blocks so if we imagine our three-dimensional axes in x y and z with those axes we can imagine taking one step so unit length or length one and calling the direction in the positive x direction calling that a vector it has magnitude and has a direction we call that i we also then come up with a name for the positive y direction unsurprisingly that gets the label j and if we also go vertically with length one we give that the label k by creating those three vectors we've actually created the possibility of generating any vector we like because we imagine going out to some point in x some point in y some point in z and connecting it back to the origin then we can achieve or get from that start point to the end point by taking our i vector and then multiplying it by some amount basically taking that many copies of the i vector so i vector i vector i vector would be three steps in the x direction and then we can do the same thing in the y direction by adding up j vectors and then the same thing vertically by adding up some combination or some multiple of the k vectors and so this is a really nice way to be able to break down coordinates or relate coordinates to vector notation once you do that a few times though we recognize very quickly that the i j and the k always represent the same things we always put them in the same order so the thing that distinguishes one vector from another is the coefficient out front this v1 v2 v3 and so most the time in this course we're going to associate the components themselves just those coefficients put them in angle brackets to indicate that we're talking about a vector specifically and we'll just put the numbers that correspond to how much in the x direction or how many i vectors how much in the y direction or how many j vectors or and how much in the vertical direction how many k vectors and that is called the component representation of a vector any vector that we can draw can be rewritten in this numerical format as you read other textbooks you may see other notations for vectors sometimes square brackets sometimes round brackets may or may not have the hat sometimes they use capital letters there's definitely some variation but at the end of the day whenever you see a list of numbers like this and it's representing a vector quantity you'll recognize that in how it's being used from the context for this course we will use either the ijk notation with a addition in between or we will use the angle brackets going forward as a quick little demonstration here we have the graphical form of vectors and we can convert those into components fairly easily v1 for instance here is two steps to the right so if i think of just talking about the i vector in this direction that would be in the positive x direction compared with the y direction would be a j vector if we treat y as vertical in this two-dimensional diagram so v1 could be two i's or we could say v1 equals two in the x direction and 0 in the y direction there's no vertical component to that vector taking a look at v 2 here we have one i and a j we can write it in the component format of one x and one y one step to the right one step up we get something a little different if we look at v3 which is that we do have a step to the right but then we have three steps down well i've already captured this a bit intuitively we can subtract three instead of going up three we can go down three we achieve that with a minus sign we can subtract 3 j's or we can write that in component form as the coefficient of the i that's the 1 and the coefficient of the j negative 3. and last but not least we can look at v4 which is backwards two so minus two i's in the x direction and then up two so plus two js again or in component form negative 2 positive 2 just taking or capturing those coefficients and putting them in the order of i and j with all of that in mind take a moment to think about these vector component representations and then think about how we could use those to help us compute vector sums take a look at v1 take a look at v3 and then imagine what you would get as the sum of those either by drawing a picture or by looking at the components pause the video for a moment and then we'll check in all right let's do this quickly in both ways if we take v1 plus v3 v3 is like this we already have v1 drawn out so v3 added on the end would look like this diagram here and so the sum would be this combined vector in green that would be v1 plus v3 well we can see that that would be 3 to the right and 3 down so we'd expect that to be 3 negative 3 which is in fact our list here as b however if we think of how we built this diagram we could do exactly the same thing going straight from the components v1 was two zero v three was one negative three what do we do when we add vectors on the plane well we take the x components that we had in the original vector the first vector and then we add in the component of x from the second vector so we take this 2 and this 1 and that gives us a 3 for the x component we then do the same thing for the y component so we have zero minus three we have then at the end a new vector the sum vector is three to the right and three down or the three negative three vector in component notation if we do that again take a moment to practice and ask yourself what v4 minus v3 might be pause the video for a moment and then we'll check in all right we have v4 which was negative 2 2 and we had v3 which was 1 negative 3. now we just have to be a bit clearer about how we handle this negative sign v4 minus v3 we can replace each of those with the components we started with now with a minus sign in between and we do exactly what you would expect we would take this negative 2 and we would subtract the matching x component in the next vector so we'd combine those two with a minus sign because of the subtraction between the vectors so we end up with negative 2 minus another one that would be negative three and we repeat the same calculation with the y components two minus minus three is going to be plus three that would be five does that make any sense absolutely if we draw that same situation out we're going to draw it down here to give itself enough room here we have v4 and then v3 would ordinarily be down like this but we're going negative v3 so we go in the opposite direction and if we do that we are going to obtain a vector that is two steps and then one more step minus three steps to the left and then in the vertical direction we would be going up two and then we'd be going up three more so we'd have that total vertical component of five once we have the component form of a vector we can also easily calculate the magnitude and this is based strictly on pythagoras imagine if we had components that was this much in the x direction this much in the y direction let's call it v1 and v2 is scalar values then the length here of the hypotenuse because this would be a right angle is v1 squared plus v2 squared all square rooted notice if we have a three-dimensional vector the length is exactly the same but then we just add in the third component squared so take a moment to consider what the length of the vector in component form to negative 3 might be among the options here we'll check in in a second all right well if our vector is 2 comma minus 3 then that in this idea here is what we call v1 and v2 the scalar values and so the length is simply the square root of 2 squared plus negative 3 all squared notice like with pythagoras we expect all the quantities to be positive by the time we're done because the square makes everything positive so we'd have the square root of 4 plus 9 or the square root of 13 answer c with that we've covered all the fundamentals of vectors as direction and magnitude quantities and how we can represent them in components and then use those components to compute interesting properties like their magnitudes