PHY 101//INTRODUCTION TO SCALAR & VECTOR// TYPES OF VECTORS

Transcript

you are welcome to the gimat 41's General Physics class in a previous class we treated the concept of dimensional analysis and it was stated that a physical quantity may have Dimension and then there are some other physical quantities that do not have Dimension we refer to them as dimensionless quantities but then we focused more on physical quantities that have Dimension and we actually did some get work in that right in this video we are going to look at Scholars and vectors our class is going to focus more on vectors however in this video first we going to St definition of scalar quantities and definition of vector quantities with their examples just like we learned from previous classes physical quantities can either be derived or what fundamental and then these derived of fundamental quantities can still be further classified as scors and vectors some quantities can be described completely only by considering their magnitude alone magnitude is the same as length is the same as modulus is the same as size those quantities that you only describe them considering the magnitude alone are known as scalar quantities and then there are others that stating their magnitude is not enough to describe them completely you need to attach something else and that something else you attach is their directions so physical quantities that are defined in terms of their magnitude and directions are known as Vector quantities having stated both scalar quantities and Vector quantities are the point there is that scalar quantities are quantities that have only magnitude is that okay in fact they are the same as numbers that's what we have on the board here that they are synonymous to numbers is that clear we have some examples of scal quantities here the list is not all so you can make a research and that and then add up to the list time is a scal quantity length mass we have temperature energy volume density speed work area and the rest of them it is possible good to carry out certain mathematical operations on scalars like adding them it's possible you can subtract scalars you can multiply Scholars you can divide Scholars does that make sense just think about an example just that we have on on the board if the gimat 41's tutorial class is meant to last for 50 minutes and then it happen that we stopped 10 minutes earlier one can obtain the time spent for the class actually from doing 50 US 10 minutes that we start earlier than normal that's going to give us 40 minutes which means that the class lasted for 40 minutes now did you see something that we've been able to subtract these scalars and then got another result which is also a scalar that's what we have here consider also a case where a train is moving with maybe covering a distance of 100 kilm and it took the train 2 hours to cover the distance one can calculate or determine the speed with which the train covered that distance in that time we know that by definition speed is distance over time is that okay so all we need to do is 100 / 2 we get 50 which means that the train sped with 50 km/ hour that is the train speed is that okay is 50 km per hour per of speed is sp anyway okay so we'll be able to Define divide these two scalars to get another result which of course happens to be a scalar as well too so actually we can add subtract and multiply scalars we can also divide them we now want to focus on Vector quantities and of course this class would really treat Vector quantities in full that's actually our focus in this class is that okay Vector quantities are those quantities that can be defined in terms of their magnitude and what direction So Physical quantities that can only be completely described considering both their magnitude and directions are known as Vector quantities examples of vector quantities we have them like displacement velocity acceleration force to moment momentum momentum can either be linear or angular momentum we have electric field in fact all our field Electric magnetic and gravitational field are vector quantities we also have polarization weight impulse current density they are all Vector quantities what about this quantity known as AB have you heard about it before ABS have you been able to integrate displacement before if you integrate displacement the result you're going to get is a vector quantity known as absement what about differentiating position with respect to time if you differentiate position with respect to time once you get velocity if you differentiate position with respect to time twice you get acceleration I'm going to ask you a question what if you differentiate position with respect to time three times what will you get you get this physical quantity which is also a vector known as jerk in the next video we're going to treat representation of vectors and then we're also going to consider the different dimensions that vectors can exist in this video we're want to look at how to represent a vector and then generally I'm going to give a summary to that a vector is simply represented by a line let's say this is the line or the line represents the length the magnitude the modulus the size of the vector and then with an arrow head let me use the cover of this marker as the Arrow Head the Arrow Head represent the direction of the vector so looking at this that the cover is facing this direction it means that this Vector is moving in this direction that is my left but to your right is that okay now the starting point of this line is called the origin of the vector we refer to it as the tail refer to it as what the tail of the vector that is where you started the line from is that clear where you started the line line from so call it the tail or the origin of the vector so that's simply how we represent Vector using the line and an arrowhead the line represent the size the length the magnitud modulus while the arrowhead represent the direction when we are dealing with Vector let's say um geometrically are you getting where one can one can construct the resultant of a vector possibly from triangular lws of vectors or parallelogram LW of vectors you know from um geometrical analysis one can construct these lines this magnitude of the vectors you draw it to scale you might choose to say okay let's let 5 cm represent one unit value of the vector so if you're given something like 15 unit value of the vector then you're going to draw there about 5 * 15 is that not true to know the length of the vector since we saying that one unit of the vectores 5 cm may be on your ruler so you can do such con construction and get an accurate value of the resultant of a vector when we talked about scalars we explained our scalars can be added subtracted multiplied or divided similarly we can carry out operations on vectors but one of them it's not possible you cannot divide a vector by another Vector you can only add subtract and multiply vectors but you can't divide a vector by another Vector in terms of division you can only divide a vector by a scalar take note of that different types of vectors there about 11 of them anyway but I'm going to mention only six and out of the six here on the BR we shall only be interested in four of them the first number two number three and then number six is that okay so types of vectors quickly let us look at parallel vectors when are vectors said to be parallel all right now look at this there is a parallel Vector if I assume the markers to represent the vectors how do we know that number one the vectors are pointing to the same direction is that okay when vectors point the same direction refer to them as what parallel vectors most cases the angle between them is equal to 0° take note all right parallel vectors may be equal or not equal for example looking at this the magnitude the length of these vectors are the same because if you check the length of the marker you know that the length are equal and then the fact that they are pointing in the same direction makes these two vectors to the be parallel and equal is that okay but let us assume that this marker is now longer let's say the red marker is longer in terms of its size it magnitude than the blue marker even if they are pointing in the same direction they will be parallel but not equal being that this one is what longer does that make sense all right so take note parallel vectors also known as cinear vectors can be equal or unequal depending on the length the magnitude of the vectors but take not that for them to even be equal in the first place even if the magnitude is the same they must point in the same direction before you say they are equal if one point in another Direction even if they are equal in magnitude they wouldn't be equal take note anti parallel vectors one the angle between them is 180° so something like this is an anti parallel Vector the two vectors point in different directions so even if their magnitude are equal are you getting me right for the magnitude to be equal it is the scalar component because magnitude only defin scar is not it is the scalar component of the vectors that are equal the vector itself will not be equal because they are anti parallel being that one is pointing in this direction the other is going other direction it means that they are not equal is that okay so anti vectors can never be equal take note even if their magnitude is the same their direction is different making them unequal you could also have the direction like this and the parallel where the Arrow Head face each other orthogonal vectors when vectors are held at 90° refect them as octogonal vectors so you can say that they have the same origin starting points the same tail but points in different Direction with the angle between them being equal to 90° such Vector is as a to Vector localized vectors n vectors in unit vectors n Vector is a vector in which the magnitude is zero unit Vector is a vector in which the magnitude is one is and unit vector actually we call it Direction Vector as well too it is used to specify directions of vectors the next video we're now going to start with the algebra of vectors in one in two and in three dimensions we now going to look at the algebra of vectors okay but before we move on fully into the algebra of vectors talking about addition subtraction multiplication of vectors want to consider the different dimensions in which a vector can exist a vector can exist either in one dimension in two Dimensions or in three dimensions when a vector is given in one dimension it's simply a vector represented with a line is that okay just consider maybe an end an end that moves maybe it's climbing a tree moves up to a point all right and then turn back to the original Point again move again backward to a particular point it keeps moving along that axis it's a onedimensional vector analysis case is that okay and how we handle that is pretty simple just normal addition and subtraction for example I can say a vector of 20 Newton like force is moving in horizontal Direction say x axis and Vector of 30 newton is moving along that x axis as well to find the resultant Vector it's going to be 20 + 30 because both of them are moving in the same what direction you add them up but let's assume that one of them is moving towards the positive x axis maybe that Vector of 20 Newton positive x axis and then that of 30 newon is moving X AIS and you have to find the resultant Vector because the two vectors are moving in opposite direction you simply subtract and when you're dealing with subtraction of vector something you want to know all in all is actually addition you know the direction of vectors matter if one is moving in this direction you can take it as positive and then the other moving in the opposite direction you take it as what negative anti parallel vectors what we saw in our previous video where we talk about anti parallel vectors A and B if a is moving in this direction and B is moving in the other direction if they have equal magnitude we can say that a is equal to minus B now how does the concept of that minus come it's all about differences in Direction all we all we do in vector vector in one dimension is simply to add them up so going back to our 30 Newton and 20 Newton example if the 30 Newton Vector is moving this way the 30 new Vector if it's going this way and the 20 new Vector is moving this way this positive x axis we know that and then this is X adding both of them it means that this 20 if I take this t as positive this 20 will be what negative and so all we have to do there is simply 30 + open bracket minus 20 30 plus you always add the only thing is if it's meant to be subtraction it simply means that the other Vector is in the opposite direction to the first so it now become plus and minus giving you 30 - 20 which is 10 but if they moving in the same direction you add both of them will be positive positive which increases the value of the resultant it's all right so that's just the concept of one dimensional Vector all right that's the con I want to know the addition of vector has certain properties of Interest they are commutative that is if you do a plus b you should get B plus a is a commutative property of vectors they are also associative if you have a plus plus b in bracket then plus C you're still going to get the same result as a + b plus C I know like here you can use the low or the principle of Bard Mass finish the the bracket first before you add to the term outside so if you add a and b first before adding to C is still the same result as adding B and C first before adding towards a another principle or law or property of vector addition is the distributive property if you have a scalar times a Vector that scar will multiply each of the vectors look at this K * Vector a plus b Vector A and B are being out you're multiplying by the scar k k will multiply a and multiply B then you add both of them so you can see that K * a plus K * B then sums together now we want to talk about vectors in two Dimensions all right now when dealing with vectors in two Dimensions it means that the vector is in a plane the plane can be X and Y plane x and z plane or Y Z plane but in most cases we use XY plane is that okay the XY plane so you can see that you have the vector here is Vector op can see the length op defined by the coordinate X comma y just like we may have heard from our basic knowledge of old level when dealing with vectors in two Dimensions we can use certain laws like laws of parallelogram then triangular law of vectors for parallelogram L of vectors represent the two vectors with the adjacent side of a par but one thing is it's not usually easy to handle the the the algebra there because by the time you represent the signs of a parallelogram with vectors and you want to start solving you need certain trigonometric laws or principles like the sign rule the cosign Rule and the process could make the analysis tough is that okay using all these trick identities as the rest of the case might be so for that reason in this course we're going to focus on how to use the component of vectors to analyze vectors in two dimensions and that is how we have this like this Vector in X we use the unit Vector I to represent it then in y we use unit Vector J to represent you need to take note of this place I goes in x i is the unit Vector in X Direction and then J is a unit Vector in what in y direction we still going to make the analysis of this in the next video the formul we need here for calculations and the rest of them threedimensional Vector is analy similar to two dimensional vectors now all we need to know is that threedimensional vectors is a space Vector a vector in space so it now introduces the third coordinate which is the the Z coordinate and the Z coordinate def the altitude the height of the vector right following the analysis in terms of how to add subtract and multiply is similar to that of two dimensional vectors and that is actually what we're going to look at in next video where we treat the component of two and threedimensional vectors as well as Vector resolution in this video we want to do some analysis on two dimensional vectors remember we're talking about the algebra of vectors I want to focus more on two dimensional vectors and threedimensional vectors all right in this course is that okay now look at this diagam here is a vector in two Dimension why did we say a vector in two Dimension the point P drawn to to the O region represent Vector op and the position of this p is located using X and Y this really needs the knowledge of mathematics when dealing with two dimensional vectors the first thing of interest is to determine the displacement Vector also known as the position vector and how do we get it it's a simple one all right from the mathematical analysis and everything to combine the x coordinates of this Vector R and the y coordinate of this Vector R because we replace Op with R you know replace Op with the vector R and we calling Vector R the displacement vector or the position Vector so for us to combine the components of this displacement Vector the components are the X and Y component for us to combine it to give us one thing all we need to do is to attach J attach J to Y where J is the direction Vector the unit Vector in y AIS and then I to x i is a unit Vector in the x axis so if we attach I to x y to J it will allow us to add the two of them together and that is what we did here to actually obtain the position Vector the displacement Vector R and it's equal to what x i plus what y j so somebody can tell you that if a vector is located with a position maybe 2A 3 what is the displacement Vector what is the position Vector of the vector that has this component the first component is the X component component the second component is the Y component so the vector can whatever symbol you want to use you want to use a is your choice B is your choice I'm using R this time around is thatat my choice so the vector R in this case the position Vector R will simply be equal to 2 I + 3 J 2 i+ what 3 J as simple as so that's how we combine these vectors now these numbers 2 and 3 are known as scalar component of the vector R if you use a to represent the vector there will be scalar component of the vector a what actually made this numbers vectors is this I and what J remember numbers alone are scalars is that not so so introducing I and J which are unit vectors which are Direction vectors convert them to full Vector form in terms of numbers like this refer to them as scar component of the vector R is that okay anyway our interest is to see how we combine the x and y coordinate to get what the position or this Vector so that gives us equation one the next thing of Interest we want to see here is how to get the magnitude of a given Vector remember magnitude is the same as what modulus length or what size so how do we get the magnitude of a vector simply apply Pythagoras what theorem remember that in this diagram look at this diagram this diagram if I pull out this point like this this shape what I'm going to get is a right angle triangle don't mind what I drew here all right it's a right angle triangle where this is your R Vector R this is op okay and then this is Theta this is the Y component this is the X J and what I so you see by applying Pythagoras Theorem r² will be equal to y j 2 + x i 2 and by the time you square your I you square your J are you getting you're going to get one that to obtain your R is going to be r² = what y let's keep the I on first for now now to get your R now what do you do from here to take square root of both side you notice that R will be equal to square root of the X component squ and the Y component Square so that's how we calculate the magnitude of vector equation two here the next is the direction the direction is simply obtained from tan Theta considering trigonometric identity to get our angle Theta we know the Y component already we know our X component already so tan Theta is going to be equal to opposite over adjacent and the opposite of this right angle triangle is the side facing the given angle Theta so opposite will be Y what x to get Theta now to stand alone or subject of formula you take tan inverse of both Sid taking tan inverse of this is tan will cancel out Theta will be left then take T inverse of the right hand side so look at it how we get the direction of the vector is that clear this direction of the vector Theta is always measured from positive X AIS so even if the vector op lies in this Axis or here or here always measure the angle from positive X AIS down to the line of the vector when we trat Vector resolution you would see how important this statement I just made is is that okay all right the next thing is how to get the unit Vector associated with the direction of the displacement Vector R and the formula is simply displacement Vector divided by the magnitude of the displacement Vector so that's equation four there all these things done here is an analysis based on the cartisian coordinate or the rectangular coordinate of vector R you about caran or rectangular coordinate that's you use all the X Y as a component we now want to move on to the polar coordinate of this displacement Vector R look at this on the board the displacement vector when you're dealing with polar coordinates you are interested in two major parameters what are the two major parameters one of it is the magnitude of the vector R the magnitude and the other is the angle Theta that it makes with a positive X AIS these two things are what defines the polar coordinate of any quantity is that okay so how do we obtain the polar coordinate of this displacement Vector R is a simple one once again from trigonometric identity we can actually get that first of all is the fact that the X component of this Vector in terms of the polar coordinate is the magnitude of that Vector times cost of the angle we are seeing x coordinate in this case because X here is the adjacent are you following the opposite is y x is the part which does not face the given angle so it is resolved using cos Theta then the the vertical component is facing the given angle is resolved using sin Theta so you can see that component of the the X component of vector R is equal to magnitude of vector R cos Theta the Y component of vector R is equal to magnitude of vector R sin Theta so once we know the X component and the Y component what stops US of fixing them into equation one remove this X and put the RX value which is this and then remove this Y and fix this value because this is the Y component you have of R thet when you fix it here the equation you're going to get will give you this and then if you look at this you notice that magnitude of R is common to both of them this one and this one is common to the COS Theta part and the sin Theta part so you can factorize you can bring it out and if you factorize you get this equation five so equation five defines the polar coordinate the polar coordinate all right or the polar relationship of the displacement Vector the position Vector please take note like I stated earlier in dealing with the angle of a vector it's always advisable to measure the angle from positive X AIS down to the line of the vector and then you stop and take the value of that angle that is the angle of vector it helps us to avoid any complexity in thinking that maybe in the first quadrant s cos and tan are positive in the second quadrant only s is positive tan and cos are negative in the third quadrant only tan is positive s and cos are negative in the fourth quadrant only cos is positive tan and S are negative by the time you measure the angle of a vector from positive X AIS down to the line of the vector you don't need all those stress off here is positive this is you don't need that when we meet Vector resolution fully we are going to still repeat this and remind ourself of it as we use it now in the next video we are going to introduce vectors in three dimension be very f about that since we've made much of the analysis from vectors in two dimensions on the board here we considering vectors in three dimension the analysis we need there all right as you can see there are three equations of interest to us here and U we're going to look at that one of it is the displacement or position Vector for a space Vector that is a vector in three dimension already St that Vector in three dimension is a vector in space defined by how many coordinates three one the x coordinate the y coordinate and the Zed coordinate did you take note of the unit vectors Associated or maybe attached to each of the axis so the new thing is K which goes in the direction of what Z axis the displacement Vector is given us this displacement Vector R is equal to x i + y j+ z k if you look at this is similar to that of the two Di di Vector that was analyzed in our previous video the only differ here is that you need to introduce the third coordinate the Z coordinate magnitude of the displacement or position Vector is still similar to what we saw the magnitude for two dimensional Vector we use Pythagoras Theorem analysis so the magnitude of the vector all right we put that the magnitude is equal to otk of X2 + Y2 + Z 2 once again a new term was introduced which is z s because you're dealing with threedimensional vectors then the unit Vector associated in the direction of this Vector R for three dimension has the same formula as what we saw for two dimensional Vector once again the only difference that this magnitude here contains three AIS so that's just the only thing is that okay please note note the two or three coordinates three coordinates are mut perpendicular hence unit Vector unit Vector I comma J and K are octogonal are octogonal of course you know what we mean by saying that the octogonal Theta is equal to 90 degrees is that not so so take of that concept please because we're going to use it to analyze you need Vector multiplication under dot product also known as scalar multiplication of vectors as well as cross product also known as Vector multiplication of vectors is that okay so going to trck that take notes of the fact that these three vectors i k are octogonal at 90° we are now set to go fully into the algebra of vectors whether in two or in three dimensions when we're going to use the component we're going to consider addition so subtraction and multiplications of the vectors