Transcript for:
3D Vectors in Engineering Statics

hello everyone and welcome back to another engineering statics lecture video i hope you guys are all doing well and are ready to learn perhaps what first years hate the most three-dimensional vectors now in the previous three videos or week one videos we spent a lot of time discussing two dimensional vectors we came out with some nice formulas for things like magnitude and the unit vector and we concluded that 2d vectors they're actually pretty nice because they're very easy to look at and visualize if i had a vector in two dimensions i can very easily find the angle with the x as x axis and as against the as is but of course that doesn't make any sense and we can figure out things like magnitude force components it was all pretty simple now i kind of alluded to this in the last video that the next logical step was to take what we learned in two dimensions and apply them to three dimensions so that's what we're going to be doing today now for those you're a little bit concerned saying ah clayton you know i was very comfortable with 2d it's something we covered in high school it's something nice to me and i'm a little bit scared at 3d well don't be scared as we're going to see it's very simple and we actually don't have to do a lot we just add one minor modification to everything that we do so let's get started so the extension from 2d to 3d may sound complex but it's actually very simple because we're only doing one thing and that is adding a component to our cartesian vectors so remember in 2d if i had a force vector we said that this can be split into two components f x and f y and i can write the vector as such where my force vector is simply f x in the i direction plus f y in the j direction in three dimensions it's actually going to be the exact same thing this three dimensional vector can be split into three components that are parallel to each axis so i'd have fy fx and of course fz and you guys may have guessed writing this is just as simple it's going to be the same format as before but the only difference is at the end we add that third component so what we'd say is we have fc in the k direction so the k that's going to be our third direction if you will and fc is going to be the components of our force in that direction now the first thing that comes to 3d that students start to hate is something called the right hand rule because typically what's going to happen in exam scenarios that's a little trick for you guys is that professors will label the x-axis and the y-axis but they won't label the positive z-axis now us as engineers we always think oh positive is always up right so we're thinking oh we're good to go it's always going upwards but actually the positive z direction follows something called the right hand rule now you guys may be saying clayton what the hell does that mean well let's go over it it's a series of steps so if i want to find the positive z axis or if i have let's say the x and z axis i can find the positive y axis all i'm going to do is point my fingers in the direction the positive x-axis i'm going to then curl my fingers towards the positive y-axis and whatever way my thumb is pointing is actually the direction of the positive z-axis now you guys may be saying clayton that was garbage i don't understand what you said at all can you give me an example well yes of course that's the best way to learn these things so let's say that in an exam scenario you're given again the positive x-axis and the positive y-axis now again by default we always assume that the positive z-axis is going to be upwards but if we follow the right-hand rule in this scenario so again first step i'm going to take my four fingers of my right hand so make sure this is your right hand it's called the right hand rule for a reason and i'm going to point my fingers at the positive x-axis so it's kind of to the side here now the second step is i want to curl my fingers towards the y-axis so the y-axis of course is going this way so i want to curl my fingers now notice that if i want to curl my fingers the only logical way to do that is if my thumb is down so in this particular case the z-axis is actually going downwards because if we look i'm curling my fingers and my act and my thumb is pointing downwards notice that i can point my fingers at the x-axis but i can't curl this way so this is why it's not going upwards the only way i can curl is if my thumb is going downwards so in this particular scenario the positive z-axis is actually going downwards now let's look at the opposite scenario in this case we flipped where the x and y axis are so in this case i'd be pointing towards the x and i'm able to curl to the y axis and as we can see my thumb is now sticking up so again i'm going to curl this way my thumb is sticking up so therefore the positive z-axis in this case is actually upwards now i'm going to give you that kind of full transparency i like to tell you the trip tick tock tips and tricks i don't know why i couldn't say that two t's must be a little bit hard for me today i guess i've very rarely seen this in an exam it's one of those little things where it's fun to know but rarely have i seen you guys test it on so be aware of it but if you guys are really struggling don't be too scared something i've very rarely seen on exams so let's talk about the other two things so remember in 2d vectors we said all right we had our cartesian vector form and with that we were able to do two things the first one is the magnitude and then the second one is the unit vector what we said in 2d the magnitude was very simple because if i have the two components f x and f y well the magnitude is just going to be through simple trigonometry so the magnitude there is simply going to be f x squared plus f y squared added together and then square rooted because we have a nice right triangle here everything works out very nicely well how about in three dimensions it actually follows the same logic which is hard to believe because again as students you guys are always believing that 3d is somewhat much more difficult but it's actually not so that's the thing we have going for it it looks like but it's actually really nice so if we have a 3d vector like this what we can actually do is we can start creating a series of triangles so the first triangle i'm going to look at is in that x y plane right there if we have a triangle in the x y plane it has a hypotenuse which i'm going to call f x y because again it's in the x y plane if we were to do some basic trig on this triangle remember it's a right triangle we can figure out that f x y squared is equal to f x squared plus f y squared the hypotenuse squared is equal to the two side lengths squared and added together so so far so good nothing too crazy but what else is nice is we can form a second triangle this one going from the x y plane vertically and again this is also going to be a right triangle and we can say that for this one the magnitude of the force squared is going to be equal to f x y squared plus f z squared now it looks pretty complex but remember that f x y we actually have a formula for it it's above so from there i can substitute what i got for triangle one into triangle two and i can conclude the following where the magnitude of f is equal to the square root of f x squared plus f y squared plus f z squared so if we compare this to the 2d case all we did was add that extra component at the end that fz squared and this is where 3d actually isn't too bad because all we're going to be doing for every scenario is just adding that extra components and a little fun fact for you guys you don't need to know it for this course but if i were to extend this to four dimensions five dimensions all i would be doing to this formula is just tacking on that additional uh component squared you guys will learn more about this in linear algebra which i guarantee most of you will not like linear algebras it's it's a different type of course it's it's a lot of fun it's very useful but it's uh yeah it's different i'll just say that so we talked about magnitude we said uh easy peasy all we have to do is add that extra component what about the unit vectors clayton surely the unit vectors must have something a little different well before we talk about the formula let's just remind ourselves what a unit vector is remember in essence a vector is a magnitude which we discussed in the previous slide multiplied by a direction and in two dimensions we said that that direction can be expressed as a vector which we call the unit vector so if this was my force vector in 3d the unit vector is going to look something like this it has a magnitude of one so it's going to be smaller than the force vector and it has the same direction as that force vector so there's the key it has the same direction and again we use this to define the direction of that force vector now the unit vector is actually really nice because all we have to do to the formula compared to 2d is we're just going to add in that additional component so remember in two dimensions we said that we can find this unit vector by simply dividing each one of the components by the magnitude of the force and that same definition applies here where all i did was i took fx and divided by the magnitude i took fy divided by the magnitude and now for three dimensions i took fz and also divided by the magnitude so again same formula as 2d we just have to add that additional component now unit vectors in 3d they maintain all those same properties in 2d that we had so if you guys are thinking about unit vectors make sure you remember these three things the first one is they're just used to define direction that's it they don't have units nothing like that they solely define a direction which goes into the second point they're unitless so it's one of those things if your professor wants to be an ass and you accidentally put units of let's say meters or feet on a unit vector well you're probably going to get some marks taken off because unit vectors they're unitless they just define the direction now the last one which is very important and we've talked about this already is that they actually have a magnitude of one and you guys are saying clayton you keep saying that you say it's important but we've never had to use that idea well in this lecture later on as we're going to see this is going to become very important not only now but also in later topics so always remember unit vectors they have a magnitude of one gonna come in handy so you guys are saying well you know what clayton 3d you're right it isn't too bad all we did was add that additional component when we talked about our force in cvn we just added the k components when we talked about our magnitude we just added that fc component squared when we talked about a unit vector we just added that k component nice and simple right well yes and no so 3d starts to get a little bit more complex when we start talking about coordinate direction angles so remember in 2d if we had the two components it was very easy to see the angle that the vector made with the x-axis and if i knew the angle the vector made with the x-axis it's very easy to determine what the the angle the vector makes with the y axis i just go 90 degrees minus that angle in three dimensions it gets a little bit more tricky because we actually have three axis we need to consider so coordinate direction angles are the angles a 3d vector makes with the positive axis alright so there's going to be the key here these are the angles it makes with a positive axis so let's say i had my force vector it's looking sexy it has all the components f x f y and f said while the coordinate direction angles are going to look something like this so from the positive x-axis to our force vector we call this angle alpha from the positive y-axis to our vector we call this angle beta and then finally from the positive z-axis to our vector we call this angle gamma all right and you guys are saying clayton oh i don't like the looks of that 3d angles kind of gross but here's a little secret coordinate direction angles are actually your best friend they make everything as easy as it can be the reason why well we can figure out force components using these coordinate direction angles remember that typically we're given the magnitude of a force we're not given the force in cvn we're given the magnitude some trigonometry stuff and we're expected to figure out what the force is in cbn if we're given coordinate direction angles we can very easily find each one of the components because all we're going to do is take that magnitude which we're given and multiply it by the cosine of the coordinate direction angle so for f x i'm going to take my magnitude f and i'm going to multiply it by cosine of alpha for y same thing except cosine of beta and then for z same thing except cosine of gamma so notice one thing here these are all cosines and this will be explained right here so a couple of little things these angles are always going to be between 0 degrees and 180 degrees all right we can't have more than 180 degrees because then it starts measuring from the negative axis or the opposite way so our angles are always going to be between 0 and 180 and this is important because this allows for cosines to be positive or negative so let's say that i had an axis that is positive going to the left all right or i guess this is to the right i get a little bit confused the positive going to the right but my force vector is actually going to the left well as we can see here the angle is going to be greater than 90 degrees so let's say it's 120 degrees if i cosine 120 degrees i'm going to get a negative value so this math right here actually takes into the count direction because if my uh force is greater than 90 degrees it'll automatically become negative which indicates it's going in the opposite direction so it's actually really nice for us because this allows for those negative components as we're going to see here a lot of the intuitiveness that we've had in 2d gets erased in 3d because the math actually starts accounting for everything we need to know which is great means that we have to think less and that's always the best solution so coordinate direction angles now coordinate direction angles actually do something very special with the unit vector and this is trick number one professors love to throw at you guys at a midterm exam or even an assignment but they usually like to save it to the exam if they give it to you in the assignment then you guys know the trick if they spring it on you in the exam well you're kind of screwed so this is trick number one coordinate direction angles form a very special relationship with unit vectors so if we look at our unit vector formula it's very simple we just take each component and divide it by the magnitude of our force now we're saying clayton components we actually know what those are from the coordinate direction angles we know that f x is going to be the magnitude multiplied by cosine alpha f y is f multiplied by cosine beta and fc of course is going to be f multiplied by cosine gamma so if i take those identities from coordinate direction angles and substitute them into the unit vector formula i get the following now it's nice here because i'm seeing a lot of magnitudes of f on the top and the bottom so i can actually cancel those all out if i'm taking something and dividing it by the same thing of course it cancels out so this formula can actually be simplified into the following where the unit vector can actually be directly obtained just by going cosine of alpha in the i direction cosine of beta in the j direction and cosine of gamma in the k direction i didn't need the force magnitude at all i already know what my unit vector is going to be which is great so there's kind of the first little thing now here's where the trick comes in we know that the magnitude of a unit vector is 1. i've been saying that all the time this is where it comes in handy for the first time so if i look at that formula i have and i know the magnitude of that bad boy must be 1 well i can conclude the following that the square root of all the components squared so cosine squared alpha cosine squared beta and cosine squared gamma if i know that that is equal to 1 i can simplify it into the following all i do is square both sides the square root vanishes on the left one squared well that's still going to be one i get the following identity where cosine of alpha squared plus cosine of beta squared plus cosine of gamma squared that's all equal to one so this is where the trick comes in because what professors love to do is they only give you two coordinate direction angles and a lot of students will say well i got two coordinate direction angles i can't do anything i need that third one to actually make things meaningful well if you know two of the angles you can actually solve for the third one using this formula so this is the first trick for instance if i know what alpha and gamma is my only unknown in this equation is going to be beta so i can sulfur beta directly i'm laughing i'm in the exam they throw the trick at me but i know what the trick is i'm having a great time so that's going to be the first trick now the last thing we have to discuss is solving these vector components we went through the formula of magnitude unit vector and we initially had all the components we said we have f x f y and f z and if i know those three components everything's easy i just substitute everything into formulas but remember typically in these type of questions we're not given the components we're actually given the magnitude of the force and some sort of trigonometry and as an initial step we have to take what we know and find that force vector in cartesian vector form so for 3d scenarios we have three general cases to find these force components and it's it's not official three cases it's it's my own personal three cases so if you guys know these three cases you guys will be good to go no matter what scenario the first one is what i call the trade case the trigonometry case this is the one students i think hate the most because again it's trigonometry no one likes trigonometry so this is the case where they give us our force vector in 3d and they give us two things two trigonometric identities they give us the angle from the x y plane up until the vector so that would be theta one in this picture and then they give us the angle uh in the x y plane that theta two so in this particular case it looks really complex but it's actually just a matter of solving two right triangles so the first triangle is going to come right here so if i were to go from the x y plane out we have f x y and then if i were to go from that point upwards it would be f z so this is going to be our first triangle and the angle of course is going to be uh theta one so from here i can conclude two things my fz component is simply going to be the magnitude multiplied by sine theta 1. so that's pretty easy right off the bat i already have one of my components now if i were to do cosine i'm not getting fy or fz i'm actually getting f x y so this would be the vector in the x y plane and again that's just taking the magnitude of f and multiplying it by cosine of theta one so this point i know one out of the three uh components but what about the other two well these can be obtained by using the second angle theta 2 and again creating another triangle so i can go fx in the x direction and then f y in the y direction and as we can see it forms another triangle with f x y so in this case my f x is going to be f x y times cosine of theta and my f y is going to be f x y times sine of theta two so that f x y even though it seems like a useless intermediate step we actually need it to solve for f x as well as f y at the end this is the worst case but it's actually pretty simple it's just four calculations you'll be good to go you'll know all of your components and then you can proceed to things like the unit vector magnitude stuff like that now the second case is what i call the coordinate angles case or the coordinate direction angles and this is the one we just talked about it's going to be the easier case so this is when they give you a force vector in 3d and they're very generous and then they give you the three coordinate direction angles if this is the case we said clayton this is a joke we know that we can find those components if we know those coordinate direction angles f x is simply going to be the magnitude of f multiplied by cosine alpha f y is going to be the magnitude of f multiplied by cosine beta and finally f z is going to be the magnitude of f multiplied by cosine gamma so as we can see this is actually the easier the cases now remember the only real trick that they can give you in this one is they give you two of the coordinate direction angles let's say alpha and beta and then you're saying okay well how do i find gamma well remember that cosine alpha squared plus beta cosine beta squared plus cosine gamma squared that's all equal to one so if we know two of them we can solve for that third angle and once we know all three we can figure out every component so these are going to be the two kind of main cases you're going to see initially now you guys are saying clayton you idiot you have a typo you said that there's three cases but you just showed us two well there actually is a third case but it's not going to be in this video because the third case is much different much more different than these first two cases and it uses something called position vectors which are going to be extremely important moving forward the reason why i'm not discussing in this video is because it's not like the first two cases the first two cases involve trigonometry every single calculation here is cosine or sine of an angle now this is impractical in reality let's say that i have a giant wire that stretches to a telephone pole provides it some support as an engineer you're not going to go in the field with your dinky little protractor and start measuring angles you you you'd look like an idiot so these two cases they're good to know but in reality they're impractical because we don't go out into real life and start measuring angles of things well we know our position our coordinate points we would know the location of something and we would know the location of another thing as we're going to see in the next lecture if we know the two locations we can actually solve for components really simply so that's a little uh oh i'm alluding to the next video so it's great so for now we need to know these two cases and then we'll discuss the third case in the next video so now that we know the components of a 3d vector as well as the formulas for the unit vector magnitude and coordinate direction angles the last thing that these questions really ask us to do is just add the vectors together they'll give us maybe three vectors and they'll say you know what i want that resultant vector well if we have all of our vectors into cartesian vector notation we can add them using the same process as before where the x component is simply going to be the summation of all the x components the resultant y component is going to be the summation of all the y components the only thing that we're doing is we're now adding those z components but again we just take them we add them all together best way to show this is an example so let's say that they give you three vectors and they say you know what clayton i want that resultant vector f1 plus f2 plus f3 and you guys may be saying oh this looks like a lot of work actually it's a piece of cake fr or the resultant vector is going to have a component in the i direction so i'm going to do is i'm going to take the i components of f1 f2 and f3 combine them together and then i get 2 minus 1 plus 3 and that is going to be my resultant i component for the j component again all i'm doing is i'm taking all three of those j components of f1 f2 and f3 adding them together and then the same thing for the k component so my resultant force is going to be 4i plus 2j plus 3k and now that i know this i can figure out the unit vector i can figure out the magnitude all of that fun stuff so as we can see it's actually looking very nice now you guys are saying clayton is it's looking nice but i'm not stupid it's looking too nice that's what you kind of learn in your later years is something looks nice while there's probably something bad around the corner and the answer is yes and no so that's probably not the answer you guys want to hear but that is this thus far we've always just said that if we want a resultant vector just add everything together you're good to go and that's true because up until this point we've dealt with something called concurrent systems and next week too we're still going to be dealing with them so it's okay concurrent systems are fine but what happens is all these systems or all these forces act at the same point and this allows for all of that simplicity this allows us to take all of our forces add them together because if i take everything and i put it at the same point this is my airpods case well as we can see this thing is just going to move nicely all right it remains a point but as we're going to see in week 4 and beyond we're going to deal with something called non-concurrent systems and this is when our forces do not act at the same point so it's going to be something like this and this is where the complexity starts to happen because if i have a force acting at the ear and another force acting at the ass and i were to push as we can see this thing starts to rotate and that's where things are going to get a little bit more complex so don't be alarmed that it's really easy right now because it's going to get a little bit more complex in the future but overall it's actually not going to be too bad so yeah that's it for this video i hope you guys are nice and comfortable with 3d vectors again sounds like garbage but all we're doing is adding that extra component in the next video we're going to be discussing those position vectors which is our third case defining vector components and then after that i'm going to show you guys the first real examination trick all right so there's there's going to be two lectures that are before the midterm that always gets students and that's going to be that video coming up and then there's going to be one later on but we'll save that a piece of trash for later because it's probably the worst out of all engineering statics so yep that's it for this video i want to thank you guys so much for listening i really appreciate it i hope you guys have a wonderful day and i will see you in the next video