hello everybody and welcome back to another lecture video in this lecture video we're going to be talking about vector addition using the parallelogram method so kind of the idea that i have it in brackets parallelogram method will lead you to believe that there is more than one method and as you guys will see there are more than one method to add vectors together but we're going to start off with the parallelogram method so this is where the real meat of engineering starts is vector additions can be the first real topic that may puzzle a couple of you students but don't worry we're going to go through it together so with that being said let's jump into it so as we kind of mentioned last time vectors are a little bit different and then they follow specific mathematical principles for multiplication addition subtraction etc now in the last video we covered multiplication by a scalar and as we said it's actually pretty simple we just scale the vector accordingly and maybe flip the direction if the scalar is negative now when it comes to addition and subtraction it's actually a little bit harder than multiplication by scalar because as we mentioned vectors have direction and this direction must be accounted for so vector addition in essence is going to follow the parallelogram law of addition you're saying clayton well what exactly does this mean well let's say that we have two vectors we have vector a and we have vector b and we want to add those bad boys together so what i want to do is i want to create a vector c which is vector a plus vector b well what i would do is i would take vector a place it down and then i would take vector b and place it on top of vector a and the resultant vector c is going to be this green vector here so it's going to start at the tail of a and go to the head of b now you guys think clayton you're an idiot that's not a parallelogram that's a triangle you're right so why is it called the parallelogram method well notice how i started with vector a and then went to b because a b c d of course i would do it that way but if we were to start with vector b and place that one down first and then move on to vector a now you can see that we actually created that parallelogram so the key takeaway here that i want you guys to know is it does not matter which vector you start with you can start with a you can start with b so start with the vector you guys feel most comfortable with now to actually add these vectors and analyze them we have two possible methods the first one is the parallelogram method which we're going to be covering in this video and the second one is the cartesian vector notation method which we'll be covering in the next video so the parallelogram method directly involves solving that parallelogram that we created in the previous slide using trigonomic identities the problem however is that if we remember that parallelogram it wasn't a nice right triangle we didn't have 90 degrees anywhere so we have to go to a little bit more complex type of trigonometry which is the sine law and the cosine law i say complex but you guys are very smart kids you guys are going to say clayton are you joking this is simple it's not complex at all so let's review what sine law and cosine law are so if i were to have a triangle so again notice that this is not a right triangle again very rarely will you guys see a right triangle using the parallelogram method and if we have this triangle we can define each side length so again these are the lengths of the sides as capital a b and c so again these are the side lengths and inside of that triangle we have three angles so we have lowercase a b and c and we call these interior angles now note that i put the interior angle small a opposite from side length a so that's kind of a little fun fact to help you remember where these angles are now the first law we're going to talk about is sign law now sign law is very great because it relates a side and an interior angle to another side length and another interior angle so we're going to typically use this if we're solving for another interior angle in these types of questions where you're given two vectors to add together well you have the two vectors so you know what a capital a and capital b are going to be and using some trig you can find out what may be small a is so finding that other interior angle is actually going to be quite easy now that's sign law we also have cosine law it looks a little bit more gross but if you look at it intuitively you'll notice a pattern so it's not that hard to remember now we typically use cosine law to find side lengths and the key giveaway here is when we look at the equations we have the side length is equal to something all right the side length is equal to something so typically what we do in these problems is we solve for our resultant force c using cosine law and then we solve for the angle of c using cylon when we go through the procedure it's going to be very apparent here so that's kind of the trigonometry that we are going to use in the parallelogram method now i may have given you guys the trigonometry but the method still makes no sense so let's do a little bit of an example procedure on how exactly we use this method so let's say that you have a question and a professor says i give you vectors a and vectors b i give you an angle vector b makes with the horizon and angle vector a makes with the horizon and i want you to add these two together so this is a very typical assignment or exam type question where they give you two vectors they give you some geometry of the vectors so an angle and they say add them together well the first step that we want to do is we want to create that triangle remember we're going to utilize our trigonomic identity so we need to create a triangle so what we're going to do is we're going to take one of the vectors and we are going to place it on top of the other vector now remember what i said it doesn't matter which one you guys do so if you guys want you guys can take b place it on top of a or you guys can take a and place it on top of b for this particular one i'm going to say all right i'm going to take a and i'm going to move it at top of b so that's the first step next thing that we're going to do is we're going to draw our resultant vector to create that triangle so remember that the resultant vector is going to start at the tail of b and go all the way to the head of a so in this case our resultant vector c is going to start at the very left side and it's going to kind of go up to the right side over there and now if we look we have a nice triangle now this is where the cosine law and sine law are going to come into play because we can start solving for this triangle so the first thing that we have to do is we have to solve for an interior angle and you guys are saying clayton why do we have to do this well remember that in cosine law if we're trying to solve for the magnitude of c we need one of those interior angles in this case we need interior angle c which is going to be over on this side remember the magnitude of a and b is something you're given in the question so we almost have all the pieces to use cosine law the last thing that we need is actually going to be that interior angle c the problem is is if we look at this it's not really apparent how we find interior angle c so how do we find that interior angle c well the best way is to use the fact that we can draw a horizontal line if we look at this horizontal line we know that the angle it makes of a half circle is 180 degrees so if i know the angle on the left side of c as well as the right side well then i can figure out what c is so the question is what are those angles well if we look at the left side of c we actually know that that angle is going to be alpha so now that we know the angle on the left side we can say all right well i can figure out what c is if i know the angle on the right side well this one's actually even easier because we know that angle is simply going to be beta we're already given the angle vector a makes with the horizon down below so we can just kind of copy it to the top now so remember alpha and beta is something that we are given initially in the question so if we want to find interior angle c all we need to do is go 180 degrees minus alpha minus beta we have what our c is now since we know what c is we can actually solve for the magnitude of vector c using cosine law we have all the pieces we need remember we needed the magnitude of a the magnitude of b and that interior angle c so at this point we're good to go we know exactly the magnitude of c another thing that we have to do is we have to give some sort of indication of the direction of c remember that a vector is a magnitude which we just solved for and a direction but we haven't really specified a direction if we look initially when we're given vectors a and b they gave us the angles alpha and beta to define that direction so we need to figure out an angle to define our resultant vector c how do we do this well we're going to go to sign law and we're going to start solving for interior angles now if i'm looking at c one interior angle that might be useful is going to be this interior angle a now this is great because we can use sine law because remember in sine law if we have a length and an interior angle we can solve for the interior angle if we know the other length in this case we know the magnitude of c as well as interior angle c and the magnitude of a therefore we can easily solve for that interior angle a so this point now we have that light blue interior angle a well from there we can say all right if i know what alpha is and i know what a is well i can figure out what this angle gamma is which is the angle that c makes with the horizontal axis so in this case i would say that c is maybe 32 degrees with the x-axis so remember when we're specifying the direction of these vectors we have to give it in terms of an angle so that's basically the parallelogram method the problem is is we have a couple special cases that professors like to kind of throw at you to test your knowledge and the first one is vector subtraction so again we just talked about how we add two vectors together what happens if we want to subtract them well it's actually really simple because subtraction is a special case of addition just like how division was a special case of multiplication if i wanted to take vector a and subtract vector b to get my new result in vector c well this is the exact same as going a plus negative one times vector b now we covered multiplication by a scalar in the previous episode so we already know that what negative one does is simply flip the direction of the vector that's all it does so if i were to take these two and add them together all i need to do is flip vector b before doing the addition together so if i have vector a and vector b just like this and i want to subtract vector b from vector a well my methodology would be exactly the same just with that little trick thrown in so i would place a down something like this and if i were to add them together normally i would place a vector b like that but since i'm subtracting them what i would do is i would actually flip vector b the other way and then i can create my resultant vector c and then if we look here we have our triangle again we can solve for those interior angles and we can solve for everything that we need so again it's just a special case of addition all you guys need to do is flip the vector you're subtracting and you're good to go the last thing that we're going to talk about is the addition of three or more vectors we just talked about adding two vectors together and adding two vectors it kind of sucks because we need trigonometry but you guys are all very smart students you guys can handle it when we have three or more vectors we don't really create a triangle you're saying clayton how do we not create a triangle let's take a peek so let's say that i had three vectors a b and c and i want to add all three to get vector d well using the procedure we learned before we place a down then we place b down and then we place c down well we can see that our resultant vector d is going to look something like this well if you guys are solving it this way all i can say is good luck because if you look here we didn't create a triangle and if we didn't create a triangle we can't use sine law or cosine loss so this is not the way that we actually add three plus vectors together we do something very similar but basically we split it into steps so if we add three or more vectors all we're going to do is we're going to add two vectors at a time to create intermediate vectors which are then added to the remainder vectors saying clayton that sounds a little crazy what do you mean well if i was to solve the situation above i would place a down and then i'd place b down now at this point i have my two vectors down so i'm going to solve for an intermediate vector which i'm going to call r1 which is a plus b after that i'm going to take that intermediate vector and i'm going to add the remainder vector so in this case i've already added a and b together but i still have c left so i'm going to place c now on top of r1 and then this will give me my resultant vector d so it's just a matter of splitting it into steps and from here i can solve for our resultant vector d and that's it that is vector addition using the parallelogram method now another thing i'm going to probably mention at the end of every one of these videos is the best way to really understand this is through examples the best way to understand the theory in this course is through actual applications so if you guys look in the description below i'm going to have two example videos one on how to add two vectors together and then one of course to add three vectors together just to kind of cover everything we learned in this video here so yeah that's it for this video guys thank you guys so much for listening i will see you guys in the next video you