hello everyone welcome to our classroom thank you for taking the time to join us today we are starting a new topic vectors and as customary in this first lesson we're going to spend some time talking about the language of vectors and those little things that we are to know before we can head on into and dive deeply into the topic and so our objective objectives for today are to one describe define the term vector so describe or define and to show how vectors are represented to state the types of vectors that we'll have to be working with and to write the component form of a vector so we're going to do a little bit of math here and the last thing we're going to talk about equal and opposite vectors which are really relationships between vectors so let's start off by talking about what a vector is in mathematics vectors are a totally different thing from in biology biology has vectors as things that carry disease in mathematics though a vector is a quantity that has both size or magnitude and direction and that direction is normally an angle so there are two types of quantities in mathematics we call them scalar quantities and vector quantities and vector quantities are those things that have size and direction or sometimes we just say magnitude and direction if if a thing has magnitude and direction then we call it a vector quantity and if it has only one of those things we call it a scalar quantity so what are some examples of vectors some examples of vectors are these velocity velocity is a good example of um of of a vector the velocity as you know is speed in a particular direction so speed has a measure of size and it's going someplace so second one is gravitational pull which we call weight so you have a mass a body has mass and then gravity will pull it downward so um weight is one of those quantities that we call a vector quantity it has mass and it has direction which is done and bearing now when a ship or an airplane sits off on a bearing it travels at distance and the angle tells you where it's going so a bearing normally is referred to as a vector quantity and of course force force which is either a push or a pull those are vector quantities because you have power in the push and that which has to go in some direction so there are quantities that have both magnitude and direction or size and angle and there are some quantities that don't and those that have size and direction are called vector quantities and those that don't are called scalar quantities and examples of scalar quantities are they are here mass length area and volume these are all scalar quantities as opposed to these which are vector quantities so let's move on next talk about how vectors are represented a vector is normally represented with capital letters and an arrow above them that shows the direction that the vector is going so here we have a b and there's a arrow above the a b that shows you where the vector starts from starts from the a and it's going to be and we usually write that in terms of algebra as to what its representation is or we use a column matrix to show what this representation is so here a b starting at a going to b is equal to x plus 2y we could write it that way or om is equal to 1 5. so a vector is normally written in this way two capital letters in the name with an arrow above the top and the arrow shows us the direction where it starts and where it terminates um let's move on i want to talk about types of vectors now the main types of vectors that you'll be working with in cxe are the position vectors and the free vectors and if we look at our diagram over here let's read this first um well we can look at the diagram the this vector starts at the center of the of the of the cartesian plane here we call this point the origin and its name is o a and this vector is just here in the open space b c notice the arrow that it's going in the direction of a this one is going in the direction of c so a position vector starts at the origin zero zero and has o at the beginning of its name hence in this one its name is o a and i should promptly put the arrow above it to show that it's going from o to a so any vector that begins with an o we call that a position vector and if you look at your graph paper your cartesian plane and see it connected to the to the origin here then it's definitely called a position vector all right there's also a free vector and three vectors are not connected to the origin and hence their names do not start with o as a matter of convention so it does not start at origin and it has no o at the start of its name so bc let's put in the arrow to show that it's moving from b to c and bc is that vector which we call a free vector so for cxe purposes you will be working mainly with position vectors and three vectors um there is also a zero vector won't be using it much in cxe but it exists zero vector has no magnitude and it's somewhat of a mystery because it can have any direction at all but it definitely has no magnitude and because it has no magnitude it's called a zero vector and there is the last type of vector here which we call a unit vector and the unit vector is so called a unit vector because its magnitude or its size is one unit whenever you work it out whatever the unit is it's just one one unit of measure that is magnitude so those are the types of vectors position vectors three vectors zero vectors unit vectors um there are some other things that persons refer to as types of vectors but they are not really types of vectors they are mainly relationships between vectors so for example people mentioned to equal vector mentioned equal vectors or culinary vectors but those are mainly relationships that exist between two vectors that we describe that way position three zero and the unit are the main types of vectors that we have here and so now we're going to talk about the component form of a vector and here we're going to do a little bit of math um the component form of a vector is normally written as a column matrix in the form x y so what we're going to do right now is to look at these two vectors a b and the c d notice the arrow that is moving from a to b and it's moving from c to d so we can write the component form of the vector and the component form is usually written in the form x y um where x is the x coordinate and y is the y coordinate sometimes other letters are used to represent it but generally what it means is the x coordinate and the y coordinate as appearing on this um this graph here so if we're talking about a b and the component form of a b a b is simply to look at your graph and to see what the endpoint of the vector um um a b is we could also call this ob since it starts at o but since it's given this way let's just use it as a b and um the coordinate of this point is is um six six so we'd write that down as six six since the point b here is um six six so we can write a down and write there's a column matrix and this will be the component form of a b the component form is very very important to know because we're going to use that a lot to do a lot of calculations later on so this is very important to know how to write them second we're looking at cd and cd to find the component form of cd because it doesn't start here at o at the origin we need to um do a little bit of math with that we look at the cd is a free vector let's just write that and because it's a free vector we're going to find the end point and say cd is equal to the endpoint minus the starting point all right so what is the end point for cd in terms of coordinate the endpoint of cd so you notice the arrow it ends at d this coordinate is 11 negative two so let's write that down eleven negative two and this coordinate here the beginning the starting point which is three one so c is three one starts at three one and it ends at um at d which is eleven two you can count it as in notice that the vector is going this direction so you can count it starting from here so it's one two three four five six seven eight in the x direction and one two three down so we could write it as eight three let's just write it over here eight three or we can do the calculation and get it which is to say the end point which is 11 negative 2 minus the starting point which is 3 um three one i should mention that going going in the this direction here when you're counting is positive going in that direction is considered negative going up is positive and going down is negative and because of that we counted one two three four five six seven eight going across so that's a positive eight and we went down three so this should really be a negative three and now we can do the component form by calculation for this vector cd so the component form for cd is going to be 11 take away 3 we worked out just as we're working matrices 11 take away 3 and negative 2 take away 1 and that gives us 8 negative 3. so with our position vector when we're writing down the component form all that we need to do is just to write the end point this is the end point and that is it but with a free vector we need to do a little bit of calculation if you have it on a graph you can count it by looking at the direction that the vector is going and count the number of units that represents it or you can use this formula as to say the end point minus the starting point and do your calculation and you end up with the component form so cd would be eight negative three and finally since this is an introduction and we're just talking about the concepts and stuff that will that you'll have to deal with in vectors we're going to look at another question so here is something from cxe that says the diagram below shows two position vectors o a and ob here is o a and here is ob and we want to write o a and b in the form x y which means that we want to write down the component forms so o a looking at the graph o a has the coordinate the point a rather has the coordinates let me just write it here this coordinate is negative one three and this coordinate here is um for b is 5 1 and so to write down the component form we simply write the this coordinate in the form of a column matrix so the the component form is negative 1 three and for ob the component form is five one all right so we must use the arrow to show where it's going all done um that's a component form we can use this these to do other things and we will do that in other videos to follow but like i said this this is an introductory um lesson where we're talking about the concepts that we're gonna see so finally we talk about the other some of the other stuff um equal and opposite vectors and these are not types of vectors no but they are relationships between two vectors so two vectors vectors can be equal and they're equal if they have the same component form so here we have a b equal to 2 5 and om is equal to 2 5 that means a b is equal to om so a b equal om it's also parallel to om but that's something we're gonna be talking about later on and a b is equal to x plus two y um o m is equal to y plus two x which is the same thing even though it's turned around so a b will also be equal to om so if two vectors have the same components they are equal that's what it means to have to be equal in terms of vectors and the two vectors are opposite if when you add their components you get zero so for example a b here is equal to negative 2 5 and negative 2 negative 5 and um om is equal to 2 5 and notice that if we should add these we will get zero so this vector is the opposite of this the numbers in terms of signs also are opposite so a b this is two negative two that is two this is negative five that is five numbers are opposite to each other so that is one indication that these two are opposite vectors same here for a b and o m where a b is x plus 2y and o m is negative y minus 2x which is the total opposite of this in terms of sine also we can take a b which is this vector and a b is negative 2 5 and we can turn it around in terms of saying it was going from a to b now let's make it go from b to a and if we do that then we change the sign of the component so this was negative two becomes two this is positive five it becomes negative five and so a b and b a are opposites a b is opposite um to b a and once you add them you'll realize that negative two plus two equals zero and five plus negative five gives you zero and um that tells you that they are opposite to each other so those are the main concepts we're going to be working with also there is the idea of culinaire but we'll get to that bridge when we cross that bridge when we get there it's a relationship between vectors so look out for the other other videos that are going to come up where we're going to be talking about how to work out uh magnitude and direction how to look at vector algebra how to solve how to prove that two vectors are parallel or how to prove that three points on a straight line with vectors thank you for watching and continue working hard as you prepare for your exams