hello everybody my name is Imon welcome back to my YouTube channel last time in our kinematics and Dynamics chapter we discuss units and measurements today we're going to move into vectors and scalars so to set this up let's define vectors and scalars vectors are numbers that are going to have both a magnitude and a direction Vector quantities are going to include displacement velocity acceleration and force scalers on the other hand are numbers that have magnitudes only no Direction scalar quantities are going to include things like distance Speed Energy pressure Mass the difference between a vector and a scalar quantity can be quite pronounced when there is a non-linear path involved for example when you talk about in the course of a year the Earth is going to travel a distance of roughly 940 million kilometers however because this is a circular path that it travels the displacement of the Earth in one year is zero kilometers whoa what does what does that really mean well we're going to be able to really comprehend this once we discuss vectors and move into a discussion about distance and displacements all right now we said vectors are numbers that have magnitude and directions and vectors can be represented by arrows where the direction of the arrow indicates the direction of the vector now the length of the arrow is going to be usually proportional to the magnitude of the vector quantity common notations for a vector quantity are either an arrow or bold face I think here when I talk about Vector quantities I'll put a little arrow over the quantity or the variable that I'm talking about that is a vector now like we said they may be represented by arrows direction of the arrow indicates direction of the vector length of Arrow is usually proportional to magnitude of the vector quantity now what we want to discuss are calculations regarding vectors and we'll first start with vector addition and subtraction and then move into Vector multiplication the sum or difference of two or more vectors is called the resultant of the vectors one way to find the sum or resultant of two vectors say Vector a and Vector B is to place the tail of vector B at the tip of a so if you have some Vector a like this all right and you have another Vector B all right you would want to place it you want to place the tail of B at that tip of a right here all right and point it in the direction of wherever it is all right so tip to tail method you want to place the tail of B the vector B at the tip of a without changing either the length or direction of either arrow and in this tip to tail method the lengths of the arrows must be proportional to the magnitude of the vectors now another method for finding the resultant of several vectors involves breaking each Vector into perpendicular components so what this means in most cases is the horizontal and vertical components or in other words X and Y components so when you have vectors like this right this Vector has a magnitude of 12 and here's the direction it's pointing East and you have this other Vector also pointing in the same exact Direction in the E East or if you will in the X Direction both both of them point in the X Direction one with a magnitude of 12 and 1 with a magnitude of 20. well the addition of this is really easy you know they'll both point in the X positive X Direction all right because they're both in the positive X Direction only they only have an X component all right so we know the direction will just be in the positive X component and then the new magnitude of these two added vectors is just the sum of their magnitudes 12 plus 20 equals 32. so this new Vector this sum Vector right this resultant is just the addition of magnitudes and it's going to point in the same direction as both of these all right now what about if you have this down arrow that has a magnitude of 6 and this one in the up both in the Y directions all right here they're in opposite directions so essentially what you're doing here is a subtraction process or and why why is this well because you can flip this arrow and give it a negative six all right and then you're kind of adding 10 minus 6 which is which is positive 4. all right and so on and so forth now that's the tip to tail method if you're using that or if they're in the same direction it becomes really easy but another method for it like we said finding the resultant of several vectors it involves breaking each Vector into their components X and Y components so if we have a vector here like vector v all right here's vector v all right to to find more information about vector v like um its magnitude well we can break it into X and Y components all right and treat it like a right chain triangle to figure out V all right or if we're adding this vector v with another vector W all right we can break each Vector into its X and Y components X and Y components and then just add those X components of both vectors to each other and the Y components of each Vector to each other to find the new Vector of these the addition of these two vectors so given any vector v for example all right let's go back to this we'll we'll talk about these additions in a little bit given given this vector v all right we can find the X and Y components and the way we can do that is by drawing a right triangle with v as the hypotenuse all right and we have our X component and our y component that we've broken V down into fantastic now if Theta is the angle between V and the X component all right then what we can write is that cosine Theta is the X component over V and if and we can write sine Theta is y over V all right in other words all right in other words if we know what the X and Y component is all right we can work backwards to find what V is all right or if we know what V is all right we can figure out what the X and Y component are all right the X component is V cosine Theta and the Y component is V sine Theta all right so let's do a quick example of this all right let's pretend that V is equal to 10 meters per second all right that's what this vector v is and Theta Theta is 30 degrees let's figure out what the X and Y component of vector v is well the X component is just going to be V cosine Theta all right we know what V is that's 10 cosine Theta we know what what Theta is it's 30 degrees if you know your unit circle than cosine 30 you'll remember is just square root of 3 over 2. so now we have 10 multiplied by square root of 3 over 2. all right and this can be simplified easily to 5 square root of 3. so the X component of V is 5 square root of 3. and the Y component of vector v is V sine Theta that's 10 sine 30 degrees sine 30 degrees is just one half so you have 10 times one-half and that's e equal to five all right and don't forget the units meters per second all right so that's how we can take a vector and figure out the X and Y component and if you had two figure two vectors then you can figure out both of their X and Y components and take the X component for V and the X component of say another Vector called W and you add them up and that is the X component of your new summed up Vector and likewise with the Y component y component of vector v y component of a different Vector W that you're adding and then you can get the Y component of the new summed up Vector that way all right fantastic all right now conversely if we know X and Y we can work backwards to find V all you have to do here is use Pythagorean theorem all right if you know your X component and you know why you're what your y component all right then your hypotenuse is just V is equal to square root of x squared plus y squared all right so that's all that V is if you remember your Pythagorean theorem right a squared plus b squared equals c squared all right where C is your hypotenuse if you're trying to figure out what C is not c squared then you want to square root both sides C equals square root of a squared plus b squared this is just rewriting that where V is your hypotenuse x squared is your X component which is you can think of it like an A and um Y is your y component which you can think of as B in this formula all right so it's the same thing here all right and then you can also if you know your X and Y all right and you can also figure out Theta if you're not given that right what is Theta right here Theta is just tan inverse at y over X right the angle of the resultant Vector is going to be you're going to be able to calculate that by knowing your inverse trigonometric functions and this is the formula for that so now we have a formula to figure out Theta how to figure out V we have formulas for how to figure out the X component and the Y component all right so we're able now knowing how to do each of these how to add and subtract technically vectors so the X component all right the X component of a resultant Vector is simply the sum of the X components of the vectors being added so here if you have Vector one vector 2 Vector 3 and Vector three all right to figure out the new summed up Vector or resultant in other words all right your resultant is going to be the sum of your three vectors here V1 plus V2 plus V3 you know how to do this now you'll break each Vector into its X and Y components and the X component of the new summed up Vector or the resultant Vector is simply the sum of the X components of the vectors being added similarly right the Y component of a resultant Vector is simply the sum of the Y components of the vectors being added all right and here we have a protocol that sums up all of that to find the resultant which is to find the resultant which is your summed up Vector if you will using the components method first you want to resolve the factors to be added into their X and Y components add all the X components together to get your new X component of your sumdub vector or your resultant vector and then do the same with your y components all right then you can find the magnitude of the resultant all right using Pythagorean theorem all right and then you can find the direction of the resultant by using theta equals tan inverse y over X all right and so that's how that's the workflow of of how you want to tackle problems that involve adding vectors and then we could talk also about Vector subtractions specifically all right subtracting one vector from another can be accomplished by adding a vector with equal magnitude but opposite direction to the first Vector right so how is this mathematically expressed well you have um a minus B right a vector minus B vector and another way that this can be communicated is a plus minus B all right that makes it easier to to to work with in the same way that you worked with addition all right now as with vector addition then the X component of the resultant Vector is just the difference now of the X components uh and the Y component is just the difference of the Y components of each Vector being subtracted so it's the same as addition all right you just want to um subtract X components and Y components now instead of adding them all right so that's the main difference between vectors addition and Vector subtraction now what about multiplying vectors all right this gets a little more complicated and we have two categories under multiplying vectors we're going to talk about multiplying vectors by scalars all right and then multiplying vectors by other vectors now when a vector is multiplied by a scalar its magnitude will change and then its direction is either going to be parallel or anti-parallel to the original direction that is going to depend on what scale or value you're multiplying by all right now if Vector a all right here's our Vector a if it's multiplied by a scalar denoted n right here a new vector vector B will be created all right now Vector B is going to have a new magnitude and it may or may not have a new direction as well if n is positive if it's a positive scalar then b and a are going to have the same direction the the new Vector the the old Vector being multiplied by a scalar and that new Vector are going to have the same direction but if N is a negative scalar value then the new Vector is going to have the opposite direction to the vector a that's being multiplied by the scalar all right it'll be anti-parallel to the original Direction all right so that is multiplying vectors by scalars fantastic now what if we're multiplying vectors by other vectors all right multiplying vectors by other vectors is going to require us to discuss just a little bit of vector calculus all right in some circumstances we're going to want to be able to use Vector quantities and multiply them to generate either a third vector or a scalar by that multiplication now to generate a scalar quantity by multiplying two vectors all right we multiply the magnitude of the two vectors of interest and the cosine of the angle between them and what we obtain is a scalar quantity this is called the dot product all right so let's let's summarize those those few points that we just said you can multiply two vectors with one another through a DOT product and what you get by multiplying those two vectors in the end is a scalar value and that is what the dot product is it has this formal mathematical expression Vector a dotted with Vector B all right is equal to the absolute value of a multiplied by the absolute value of B cosine Theta where Theta is the angle between a and b and ultimately what we get out of this model this mathematical expression is a scalar value so through the dot product we can take two vectors multiply them and get a scalar value out all right now this might seem complicated but we'll reiterate it when we work with topics that require us to to to use dot products all right one example of this is work all right Works formula is a DOT product between force and displacement and we'll talk about that in the next chapter all right but the takeaway is dot product is when you multiply two vectors and get a scalar at the end and here's the mathematical expression for that and we'll see it when we talk about work fantastic in contrast we can multiply two vectors by each other and then obtain a third Vector all right so that is going to be cross product all right cross products when we multiply two vectors with each other to get a third Vector as our answer all right now when we are doing this we're going to need to determine both the magnitude and the direction of that third new Vector so how does this work to do so we're going to multiply the magnitude of the two vectors all right and the sign of the angle between the two vectors once we have the magnitude then we can use the right hand rule which we'll discuss shortly to determine its Direction all right so the cross product is multiplying two vectors to get a third Vector to figure out the magnitude of that third Vector we're going to use this formula right here all right it's going to be absolute value of a times absolute value of B sine Theta where the Theta is the angle between them that's going to give us the magnitude of this new Vector all right this third Vector how do we figure out the direction right because a vector has both a magnitude and directional to figure out the direction we use the right hand rule which I'm going to cover here shortly an example of of of of a concept that's going to require us to use cross product is going to be torque and we're going to cover that later down the line in further chapters but just keep that in mind right there's there's a reason why we're covering how to multiply two vectors with each other and the dot product versus the cross product it's because we're going to see it in the formulation of other Concepts later down the line all right so cross product multiply two vectors get a third Vector the magnitude of that Vector can be determined with this expression and the direction of that new Vector can be determined by the right hand rule now to apply the right hand rule to the cross product what you're going to want to do is you're going to place your right hand so that the fingers point in the direction of your first Vector a all right so Point your right hand in the direction of your first Vector a and then you're going to curl your fingers towards the second Vector B your thumb will then point in the direction of your resultant Vector your final new Vector all right and it's going to be perpendicular to both A and B all right so that's how we want to apply the right hand rule all right let's do an example here so that we can really really understand this all right this problem says we want to figure out the magnitude and direction of the resultant Vector from the following cross product all right C equals Vector a crossed by Vector B all right we're trying to figure out this new Vector C both the magnitude and the Direction all right now we're told what Vector a is all right let's draw out let's draw out a plot here really quickly all right here's our plot here's the X Direction here's the Y and this is the Z all right and then we can continue it this is this is positive Z this would be negative Z this is positive X this would be negative X this is positive positive y this is negative y all right fantastic all right cool now Vector a has an X component minus three but it has no y component all right so it only has a x component that's negative so if we draw Vector a it's going to look something like this all right it only has a x component of minus three so here's Vector a Vector B has no X component but it has a y component of plus four all right so it's going to have a plus four y component right here all right that's Vector B fantastic notice how this angle between them right is one just has an X component the other one just has a y component this angle between them is just 90 degrees fantastic having all this information means that we can figure out what Vector a cross Vector B is at least the magnitude right away right because it's just the absolute value of a multiplied by the absolute value of B sine Theta all right a is just minus 3 by absolute value so that's going to be plus 3 and b is plus four absolute value is just plus four sine Theta sine Theta is just one so really we're just multiplying three by four and that equals 12 and Newtons per meter right because those are the units Associated here Newtons and meters so you get that the magnitude of A cross B is 12 Newtons per meter that's the magnitude now how do we determine the direction of C well we're going to start by pointing our fingers in the direction of negative X all right and then we're going to curl our fingers towards B and what we notice is that our thumb is pointing downwards all right our thumb is pointing downwards in the negative Z Direction all right in the negative Z Direction so this is going to be 12 Newtons per meter in the negative Z Direction all right so that's how we figure out both the magnitude and the direction of our new Vector that we produce from cross product of vector A and B so our new Vector C has a magnitude of 12 Newtons per meter in the minus Z Direction fantastic and that is vectors and scalars for you all right we're going to take this information and in the next video we're going to talk about displacement and velocity so stay tuned and if you have any questions leave them down below in the comment section other than that good luck happy studying and have a beautiful beautiful day future doctors