 Next question find the point of intersection of a b and cd where a b cd are these given four points so basically these are the two points and Let's say a B C and d You have to find the point of intersection. This is 6 minus 7 0 This is 16 minus 19 minus 4 0 3 minus 6 2 minus 5 10 How do you find the point of intersection when you're dealing with 3d points? Could we find the point of coincident of the unit vectors? How will you find that? Writing one as lambda times another and then finding the unit vectors Here I would tell you one simple thing use the concept of section formula See, let's say the point P which is a point of intersection. Let me name this point as P Let's say this P divides this in the ratio of Lambda is to 1 Okay, and it divides this in the ratio of let's say Beta is to 1 Okay Now if you take the lambda is to one fact the point P will have a position vector of What will be the position vector? the position vector would be 6i minus 7j times lambda and 16i minus 19j minus 4k by lambda plus 1 my lambda looks like 11. So okay, and The same point P will if you take Cd into your picture in the same point P will have a position vector of beta times or 3j minus 6k beta plus 2i minus 5j plus 10k By a beta plus one Okay Since both represent the same thing. Can I just compare their I coordinates j coordinates and k coordinates? So if you compare their I coordinates, it's 6 lambda plus 16 by lambda plus 1 equal to This side you'll have 2 by beta plus 1. Okay, if you compare the j coordinates then Minus 19 sorry Minus 7 lambda minus 19 by lambda plus 1 is equal to j coordinates here would be 3 beta Minus 5 by beta plus 1 if you compare the k coordinates then minus 4 by lambda plus 1 will be equal to 10 beta minus 6 Or what am I doing 6 beta? minus 6 beta plus 10 by beta plus 1. Okay Now one thing that you would have noticed that There's a stock resemblance between this term and this term. Isn't it? This is just this term multiplied with a minus 2 isn't it? correct So can I say minus 2 times minus 7 lambda minus 19 by lambda plus 1 is actually minus 4 by lambda plus 1 What I did I multiplied both side with minus 2 and equated their left-hand sides Can I do that? So lambda plus 1 lambda plus 1 gone. In fact, this and this will go by a factor of 2 Correct. So minus 7 lambda minus 19 is 2 that means 7 lambda is minus 21 That means lambda is negative 3 Once you know lambda is negative 3 you can easily get by putting your value of lambda over here So coordinates of p would be nothing but minus 3 6i minus 7j plus 16i minus 19j minus 4k By minus 3 plus 1 which is minus 2. Let's simplify this So minus 18 and plus 16 will be minus 2 minus 2 by this will be i Plus 21 minus 19 is plus 2 plus 2 by minus 2 is minus j and This will be plus 2k that means 1 comma minus 1 comma 2 would be the coordinates of this point. Is that fine? clear how it works Okay The next concept which we are going to talk about is another operation on vector which we call as scalar Multiplication so when you multiply a vector a with a scalar quantity lambda what happens to this vector now if Lambda is positive There's no change in the direction of it. Okay, but if lambda is negative then the direction of a reverses if mod lambda is greater than 1 that means There is a scale up Scale up in the size and if mod lambda is less than 1 In fact, less than 1 greater than 0 then there's a scale down in the magnitude of lambda a by Factor of mod lambda. Okay. So what is the role of lambda? It just changes the direction if lambda is negative and It gets and the magnitude of it gets multiplied with the magnitude of lambda So if let's say a is this then what is minus half a So let's say a is this vector minus half a would be the same vector half its length But in a negative that in an opposite direction. Okay Because minus sign reverses the direction half means magnitude is half. There's something which you already know okay Now the use of this is in understanding a very important concept What we call as the concept of Co-linearity of vectors concept of Co-linearity of vectors. Okay, but before I start talk about co-linearity of vectors There's something which I would like to talk about is linear Combination, this is a very important concept which is very challenging to understand in the first go because Many students are not able to relate easily with this concept. So, please all of you I'll listen to this concept very very carefully when you write a vector R as M1 a1 m2 a2 M3 a3 Etc till m and an Okay, then you say where a1 a2 are vectors a1 a2 etc R vectors of course, and I didn't mention that because I've already shown them with a arrow sign on top and M1 m2 etc. These are scalar quantities Okay Then we say that R has been expressed as a linear combination R has been expressed as R is a linear combination of Is a linear combination of A1 a2 till an is this statement clear any any doubt this stage Are you happy with this statement? Yes, sir. Okay. No doubt regarding. What do we call as a linear combination of a certain system of vectors? clear clear, okay fine now Something which we call as linear dependent and linear independent That evolves from the same concept Linearly independent and linearly dependent. What is linearly dependent a set of vectors a1 a2 a3 till an is said to be linearly dependent if If a linear combination of these vectors expressed as a null vector Implies M1 m2 or if you express this as a linear vector Then your m1 m2 till your mn all not zero. Okay, I'll give you an example let's say I Plus j and let's say 3i minus 2j Okay My question is are these vectors linearly dependent? Are these vectors? Linearly dependent. Okay, so my question is are these vectors? Linearly dependent Now in order to know that We'll take this approach will say m1 m2 m3 Let's say this is a null vector. Now if I'm somehow able to show that At least one of them is non-zero. That means all are not zero not all are zero means I Just have to show a non-zero case of m1 m2 m3 at least one or two of them. Okay Then my job will be done that they are linearly dependent If all of them come out to be zero then that is what we say. They are linearly independent Which I'm going to take up next case. Okay, but let me just finish off this up so if you see this so your I component will be m1 M2 plus 3m3 That's your eye and J component will be m2 minus 2m3. Let's say I call it as J If you're equating it to zero vector means it is saying zero i zero j correct Okay, so if you compare if you compare the coefficients m1 plus m2 plus 3m3 is zero and M2 minus 2m3 is also zero Correct. So if you take any value of m3, let's say if I take m3 value as k M2 will be 2k Okay, and here m1 will be minus m2, which is minus 2k minus 3m3 Which is minus 3k, which is actually minus 5k now if I don't choose k value as zero Okay, of course 00 is a solution, but there can be non-zero values also So if I don't choose k as zero neither my m1 nor my m2 nor my m3 will be zero correct so There exists not not all zero values of m1 m2 m3 says that this can the linear Combination of these three vectors can be expressed as a null vector if 00 is the only possibility then that is what we call as linearly Independent which I am going to take up next But before I go to the next case. Is this clear enough? Could you explain that part again see in plain and simple language if One vector could be expressed in terms of the other two. They are linearly dependent simple as that For example, if I say 3i minus 2j I Can actually create 3i minus 2j by doing this three times i Or let's say two times i Let me take Let me take minus two times i plus j plus 5i So this vector is generated by Some scalar times this vector and some scalar times this vector that means other two vectors some scalar you multiply and Add them to generate of this vector that means they're dependent on each other Okay, and this is how they are dependent Are you getting my point so we say these three vectors become linearly dependent I didn't understand how you found that too and How I found this minus two and this So, yeah, I would have taken x and y and solve for x and y but that would have just taken a bit of time That's why I see what I did was this x i plus j y into i Then I made a two equations three is equal to x plus y and Minus two is equal to x so x is minus two And x is minus two means y is five. That's why I came with minus two and five. Okay, this was an easy case That's why I did not do all these things, but otherwise I would have done it okay, so the meaning of three vectors being linearly dependent is one of them can be expressed as scalar times the other two Scalar times let's say this one and scalar times the other one okay, so If c vector is linearly dependent on a vector and b vector Actually, we don't say linearly dependent on we say all of them are linearly dependent on each other Let me just correct my this this this and this are Linearly dependent then c could be expressed as some scalar times a plus some scalar times b Are you getting it? because if you do this Then at least you have ensured that One of them is a non-zero for example here m3c this m3 clearly here is minus one isn't it If you compare this m3 is clearly your minus one So not all of them are zero Even if these two become zero at least c component will not be zero Okay, unless until you know of course two of them have to be non-zero in this case If a bc are non-zero vectors are you getting my point? So just remember the simple statement If you want to find out Whether three vectors are more than three vectors. That's the n vectors are linearly dependent Then a linear combination of this this is called the linear combination If it is equated to a null vector Then at least some of this m1 m2 m3 should not be zero That is the meaning of all not zero. I think this you will understand is better when I talk about linearly Linearly dependent sorry linearly independent. Let's not talk about linearly independent Now again a set of vectors a1 a2 Till a n are said to be linearly independent a linear combination of these vectors a Linear combination of these vectors if compared to a null vector Will only result into these scalar quantities being all zero Are you getting this point a typical example of a linearly independent set of vectors is ijk These three guys are linearly independent Okay, that means if you do some m1 i M2 j m3 k and equate it to a null vector This can only be possible when your m1 is zero m2 is zero and m3 is zero There's no other way that this can be equated to a null vector Which clearly indicates the fact that these three vectors are independent of each other, right? In other words, you cannot express one in terms of the other two That means you want to express k as let's say some m1 i plus m2 j You will not be able to m1 m2 doesn't exist in this case This is not possible because they are linearly independent of each other K doesn't depend on i and j i doesn't depend on j and k j doesn't depend on i and k are you getting my point What is the meaning of linearly independent and linearly dependent? Yes, sir