hey friends my name is c and you're watching here mr eazy and welcome to a new video for igcc matson today we have to rule some examples of video for vectors which is the penultimate lesson and we'll get started with some notations but before we get into it don't forget to leave a like subscribe and bring notification buttons on any future videos and we'll get started with notations so here's how to write vectors some examples of vectors you can see on textbook or any slides is bolded letters like a or b in this case but but because it's very difficult to write in bold in like paper we usually we usually underline the vectors so for a we can write it like this a with an underline a b c d so on but if it's from one point to another to another like a to c we write like this an arrow and a to c or o to a where o stands for origin then we have to map into order length so the magnitude the notation for it will be this modulus symbol like this between the the vectors like let's say the magnitude of b will be this right so right it will be the magnitude or the length of a or the vector so the formula for that is specifically just pythagoras theorem or basically the distance formula from coordinate geometry something of it this way if we have a vector like 3 four like three four where this basically means in the x direction and this particular means in the y direction so three to the other three to the x direction which is three and four to the y direction four it's basically this is a resultant the resultant vector or basically the magnitude so you can basically use pythagoras to work out the length which is in this case it's a five so five oops so five and we have the type of vectors so we have equal vectors if a equals b then a and b have the same direction and the magnitude of a equals magnitude of b because they're both equal but if there are negative vectors if a equals negative negative b then a and b are in the opposite direction because one's going this way and one might be going this way and the magnitude of a equals magnitude of b they will be the same length because it says a equals b like minus b and for zero vectors if a is a zero vector or have zero magnitude then the magnitude of a equals zero and four the scalar multiplication of vector if b equals k a where k is a constant which is bigger than zero then a and b are in the same direction because it's basically linked to the first one equal factors and b and the magnitude of b equals magnitude of k times magnitude of a but if b equals k a where k is less than zero then a and b are in opposite direction which is basically linked to the negative factors like so and we have unit vectors and unit vectors have a magnitude of 1. if the magnitude of u is 1 then u is known as a unit vector and basically what it is is that the unit vector is in the direction of a given vector a is equal to this one right here so what it essentially is is that it's the magnitude of the vector but it has a unit of one so we'll look more into it later so the the notation for a unit vector is a with this letter right here we call it little we call it like a hat so this would be a hat it's equal to a the the vector a over the magnitude of a so a height a hat is a length of one unit so let's an example would be if a equals three four then the magnitude would be this right here it would be five right but five is not a unit vector because it has a magnitude of five and not one therefore we have to divide the the vector by the magnitude which is a five therefore it will be this right here three three over five and four over five question one think of it this way if three four three four that means the resultant magnitude would be five right but union vector is a magnitude of one that means you have to split it into five equal sections like one two three four five and divide the vectors by five to get one equal one little slot right here which is basically one unit so it will be 3i plus 4j over the magnitude which is 5. and you can write it this way too and then we have addition and subtraction of factors so here we have the triangle uh triangular law of addition basically pq plus qr equals pr where basically the q basically just cancels out and just goes from p to r and number two parallelogram law of addition if p like p q plus p s equals p r and as you can see here p q plus p s equals p r like because what you can think of is that if it's a parallelogram that means these two sites must be the same right so if this were to be ps then this has to be ps2 plus the same length same direction and same magnitude and here we have the competitive floor which basically states that a plus b equals c and b plus a plus c hence a b a plus b equals b plus a and the subtraction of vectors is basically if q are basically if qr equals pr minus pq and now we'll talk about expressing vectors so here's the vectors expressed in two in terms of two non-parallel vectors if a and b are two non-zero non-parallel vectors any two vector op in the plane containing a b can be expressed in terms of a b that is op which where o is specifically the origin to p a point equals m a plus n b where m and n are constant so if p a plus q b equals r a plus s b that means p equals r and q equals s and here we have position vectors and called linear points where co linear basically means on the same line so for position vector the point is always relative to the origin that's an important thing to note so it will always be o something o c o d or e or whatever because o means the origin so for example if the positive vector of the point a equals two p minus three q and o a equals two p minus three q so in the diagram over here the positive vector of the point p with respect to an origin o is indicated by the direction that is segment o p this one right here so if the vector o p basically o p is called the position vector of p relative to o and called linear points collinear means the points are parallel and share a common point so here we have three distinct points a b and c they are called linear meaning a b which is this system right here equals b c but it's a scalar multiple because this may be like two units this may be one unit that means we have a scale factor of one over two right then we have vectors in the cartesian plane or basically an x y plane which is what we normally use so let i and j be the unit vectors in the positive direction of the axis and the y axis respectively then the positive vector of the point p x y is lp equals x i plus y j and when we have i and j as notations we call it i j notations so here's uh if something is i plus j it's the i j notation but if it's x y if it's basically the column vector notation column so if y equal if a 7 minus 12 is a vector it means it's equal to 7 i minus 12 j so its magnitude is op equals root of x squared plus y square where it's basically in this context and the unit vector and the vector then we have the distance between two points which is basically the same as coordinate geometry we can use pythagoras theorem or the distance formula from coordinate geometry so if we were to find the distance between p and q we will have two we need to have two points one one p point and one q point or the position vector and we can use x by two minus x1 squared plus y2 minus y1 squared then we have some examples for vectors express each of the following sums as a singular vector so we can use the triangular law of addition or like the addition laws so when we have a b plus b q it becomes a q right because this basically cancels out and just goes to here but as you can see the next one q to q it's the same so you can cancel this out too that means you can basically go from a to p right because everything cancels out therefore it will be a p like so and for b it's basically the same we can cancel these out these and these therefore it'll jump from a to d therefore it's equal to a d like so so the points p q r half positive vectors a plus b three a minus b six a minus four b respectively find p q and p r so number we have p q which we can basically find by doing q like the positive vector of q minus the position vector of p because you want to go from p to q right let's say if it's this pdq you need to go from this minus this value right here to get this going here so just just let's just do it to like make it make sense so q is basically 3a minus b minus p which is a plus b a plus b which equals to 2a minus 2b minus 2b or if we factor rise to two it'll be two a minus b so to initial what i meant let's look at the point p p is a b right if we pass two a it'll be three 3a that means it's 3a and from uh q of p is b right for the b vector if we do b minus 2 b it'll be minus b which is what we have here that means p q is specifically 2 a minus 2 b and for b we have pr so p r equals specifically r minus p which basically what's r r is 6 a minus 4 b minus p minus p p is a plus b therefore if we were to simplify it it will be six minus one equals five a minus five b equals 5 a minus b and that's the answer and the last question find the magnitude of each of the following vectors and write them to the unit vector and the vector of magnitude of 65 which are in the direction of the given vector so we can find the magnitude quite easily by doing 3 squared plus 4 squared and it's a pythagorean triple so you should know this it'll be equal to five right that means the magnitude equals let's say this is a therefore the energy of a will be this and the unit vector will be basically a hat it will be the it will be basically the unit vector so the vector over the magnitude so it would be 3i plus 4j over 5. so 3i plus 4j which is a vector over 5 which is a magnitude and then just use a new color and five and find a vector of magnitude 65 which are in the direction of the given vector so 65 plus 35 a hat it's basically the the magnitude that the resultant vector which is this right here but times are 65 because this is one unit and we have to find 65 units simply 65. let's see if it is 65 times 3i plus 4j over five and if you want to multiply 65 sorry if you want to divide 65 by 5 you get an answer of 13. should be 13 3i plus 4j and that's the answer and for this one right here on the on the right we have let's sum it as b right that means the magnitude of b will be equal to 5 squared plus minus 12 squared and it's also a pythagorean triple which is equal to 13 right so let me just use my calculator double check so 5 squared plus 12 squared it was 13 root of that so the magnitude would be 13. so number two b hat or basically the unit vector of b it's equal to the it's equal to the vector of b over the magnitude of b which is equal to 5 i minus 12 j over 13 right so 5 i minus 12 j over 13 and that's answer and lastly the magnitude of 65 which is basically the same as just now will be 65 b hat equals 65 times 5 i minus 12 j over 13. so if you do 65 divided by 13 you'll get an answer of 5 right so you do 5 times 5 i minus 12 g and that's the final answer and that's it for this rules and examples we've got this video for ig 60 admits for today we look into different vectors for example unit vectors magnitude and some laws and i hope you find it useful and helpful and efficient please leave a like and subscribe and ring your notification button on any future videos and with any comments or questions you feel better about my channel on my youtube or my instagram you can comment down below and i reply to them and check out my social media in the description for example linkedin or youtube or instagram and if you need any learning resources or any teaching resources you can check out my site in the description or you can come out in your browser at www.amazing.com and i hope you find it useful and helpful and i'll see on the next video which will be questions for vectors which will be the last video for the igcse animate series which would be fun but until then stay safe and happy learning [Music] you