KCET 2024 Mathematics | Vector Algebra Oneshot | Concepts PYQs and Shortcuts

hello welcome to simplified Minds VOR usually question and concept so in this video we'll discuss completely very important so now coming to Vector it's very simple chapter first of all Bas VOR pointing to a position Vector it's showing the position position usually Z from the point for example if the point is 2A 3A 4 then it is 2 I cap 3 J cap and 4 K cap this is called as position Vector unit Vector is very very very important see here vector divid by by vector by its modulus Vector modulus of the vector is given by unit Vector how do you find modulus modul root of if it is x i + y j cap plus z k so this is nothing but root of x² + y sare + Z that's a modulus vector divid by its modulus is called as what a unit vector vector joining two point if you want to find AB Vector AB Vector points A and B AB Vector is O minus O A so AB simply x21 y y1 these kind of questions can come in see cinear vector cinear vector and then basically you know a points lying on a same same plane suppose if all the know points cinear points in the for example if this say the a vector for example if I take this as a an Vector AB Vector for example if I take this as a BC Vector for example if all lay in the same line for example other colinear Vector how do you find these kind of vectors if a vector is Lambda times a BC Vector then we call cinear vectors in fact 3dom or in simple way I can say if two vectors are parall not only colinear if they are parallel for example I have now a vector I have now B Vector if a vector is parall to B Vector if a vector is parall to B Vector then we can say a vector is always Lambda time of B Vector concept important so we can always write one vector as Lambda time of other Vector know for example 2 I 3 J cap 4 K cap 2 3 4 you multiply with two for example 2 multiply 2 68 and what I'm say basically what I mean to say is that for example let me take one page um for example so there's a vector called as you know we have Vector for example 2 I cap 3 J and 4us okay - 4 I cap and - 6 J cap minus 8 k for example if this is a vector and this is B Vector so we can always write the B Vector as Min - 2 * of a vector vector VOR this is a b Vector so B Vector has minus 2 * minus- 2 -- 3 - 6 - 6 into 4-8 so we can write B Vector something some number into a vector that means if I able to write B Vector as Lambda times of B I mean a vector or if one vector is written Lambda time of other Vector that means these two Vector are parall par you whenever there's a vector because the direction is same so since your direction same you can move the vectors also like this so it is completely all right okay so B Vector I want you to remember this is very important B Vector is equal to Lambda * of a vector if they are if they parall or cinear if they give like that you can find the answer in this way even sometime they give points points you have to find the vectors for example a point B Point C point you have to find the vector for example you have to find a vector or BC vector and points will be given if a vector is Lambda * the BC Vector that means they both have same direction that means they are on same line like that we can tell we can when they say is L B VOR we can say they par but so this is what this is very important again so let's discuss about that in questions now coming to here now basically next we have one more topic that is Direction ratios Direction cosines usually vect you can't differentiate 3D and and you know 3D and vectors mix question and so Direction ratios I'll teach here I teach in 3D and the direction ratios Direction Co so what is Direction ratios for example Vector is given for example you have Vector like 2 I cap 2 I cap 3 J cap and 4 K cap so this this whatever the coefficients 2 3 4 Direction ratios Vector 2 3 4 ratios that tells about like I I directionally what is the ratio of that Vector 2 3 2 in the direction of I it is two in direction of J is three like that it will be there Direction ratios Direction cosin and we have Direction cosin l m and n basically so basic that Vector makes an angle Al beta G okay it takes Alpha Beta gamma with respect to xaxis Y axis Z axis so cosine of the alpha cosine of the beta cosine of the gamma is called as direction cosin that is cos Alpha cos beta cos gamma so cos Alpha cos beta and cos gamma they are called as L MN and if you do L square + m sare + n Square equal to 1 then we get this is one and this is a relation between the direction cosin and Direction ratios this is there I think sometime in as for the new syllabus they say this is not there when this when you learned all these things learning this one thing will not take much time so it's very important you should know so again I keep telling every time there will be something called dependent topics independent topics sometimes in NCR they say this this is not there but complication and that's it this if you know this much let's go for next topic next section formula so whenever there's a line joining it's very famous deration so you have line joining AB so it divides in the ratio of there's a point p i mean its position Vector is r and it divides the line joining a in the ratio of M is to one so that means the if you want to find this Vector R Vector the position Vector of the point P it is opposite MB into n opposite opposite MB into n it is MB into n Vector plus M plus n this is called external this is called as internal division this external division is MB n and Al sometime they ask you find the median I think C the question they asked this question C again so find the median this is a vector this is B Vector the median so Medi divide always the line which divides opposite you know line opposite base or you can say opposite line into two equal parts as median and this is median so median a plus if you want to find the length you can find the magnitude of that Vector you will get the length if they ask you intersection of medians if I take the center of this Center of this and center of this all inter particular point we call the centroid intersection of medians if you want to find that centroid Vector position Vector then it is a vector plus b Vector plus C vector by that's a vector if you want to find the that's a vector position Vector of that that a vector plus b VOR plus C Vector this is what you should know in this topic C questions most important this is the most important this is the most important point I given comparison here so this is called scal this called Vector SC for example physics physics so when you go to physics topics so there are so many like we discuss no potential energy of a dipole in a uniform electric field potential magnetic dipole in uniform magnetic field p p cos Theta potential energy so so there two for example P dipole moment so electric dipole moment Plus Electric moment and electric field p p cos Theta so vectors but when you do dot you're going to get the answer scalar energy enery scal first chapter electric charges and Fields vect I told so many times so many times I Told You So first chapter physics electric chares and field Vector quti electric field and you know your electric potential capacit potential potential scal quantity that's why directly add it science Rel to physics also prodct there a lot of questions going to ask electric charges find the minimum energy maximum energy a costet you to put you 180° and sometime put 90° to get the minimum maximum energy all those things when ioss when I two vectors I'm getting one more also a vector for example see you have left hand flaming schol right the screw rule everything happens like that in fact question find in C they'll ask you to find moving charge magnetism only they'll ask you to find what is the magnetic field what's the direction by just knowing whether plus or minus you can eliminate the options whether it's coming out out of the paper or inwards of the paper so when I rotate a screw when I rotate a screw that means it goes inwards when rotate out screw outside it comes outwards now what I mean to say that basically Vector multi I'm going to get one more Vector which is perpendicular to both the vectors for example whenever a charge moves in uni magnetic field when a charge Moves In Uniform magnetic field under circular rotation circular you know rotation the formula r equal m QB very very important QB find how we going to find by using left hand flaming tool when whenever you know whenever the charge goes for example so whenever your electron enters perpendicular to the magnetic field and it enters per and its velocity is perpendicular to that magnetic field then it's going to experience a force which is perpendicular to both the field and the current because you have this you know you have this left hand flaming so you have something called current and then you have field you have the motion so why they all three are mutually perpendicular because if I take two vectors and rotate it I'm going to get one more Vector which is perpendicular to both the vectors that's what I'm telling you physics we learn all these things to to do physics only now real life problem Sol scal your vector product is already there in real life real life already there it is happening because when when a charge enters it's experiencing a rotational Force why because that you know centripetal force is perpendicular to that I'll cut down everything because I want to do lots of questions questions very very important it is okay you going to get a vector which is perpendicular to both the vectors so when I do cross product I'm going to get a unit Vector which is perpendicular to me this answer is a vector whose direction is perpendicular to both these vectors question Vector which is to both the vectors show the questions when ENT of the magnetic field what's the force experience because Theta becomes zero it doesn't experience any Force cross product and the next Thal 90° A cross B this is what basically your a into B mod and as I told you a cross B equal to minus B cross a then is very important for C and A do b equal to B do a this is commutative this is commutative but this is not commutative that if they give two vectors like this their dot product is this the answer if they give two vectors like this the cross prodct is this is the answer basically because it is you know clearly right angle Zer there get modulus a modulus B angle 90 there get zero when I multiply I and I I get the answer X1 X2 when I multiply this into this I into J becomes 90° therefore you won't get answer is zero I into K will becomes 90° again get answer is zero so that's why you only only get X1 X2 for example multiply y1 so if you multiply y1 so y1 into this is 90° therefore you get Zer y1 into Y2 J into J you will get the answer J into K you get zero again sorry j into K also it'll become zero you'll not get y1 Z2 that means you'll get X1 X1 X1 X2 y1 Y2 Z1 Z2 I think this is what you should know this it isal and I because it's Direction it direction is very important because when you rotate like this scre going inside like this when you come out like this the screw comes outside so when electric field goes clockwise One Direction magnetic field will be there anticlockwise opposite direction of magnetic field will be there that's exactly the cross product clear next let's go for next topic quickly this is again very very important for C projection of a vector on B Vector is a b modulus B if you want you can Al write this as what a vector do what is B by modulus B you can also write B unit Vector on can write in this way also or you can say modulus a COS Theta it's simple M I already told this many times if you if you take something with respect to Theta towards the angle it is cost Theta and away from the angle it is sin Theta we always do resolving vectors that's how it is I'm going FAS next area of a triangle areal into a b AC into a b area that's the formula and what is area of parallelogram if they give area of parogram area is equal to AC into AC for example they can say area is equal to you take two vectors and then find the cross and find the magnitude going to get and sometime they ask you diagonals also if they ask you diagonals so if this is a and b so this diagonal is a plus b a plus b Aus Vector a a these are the two sides the diagonal will be a plus b and then a minus B that's it so if you remember this let's go to question quickly be very veryy because even simplified we do that concept cap video concept very small but if you see DP is all very lengthy one one one and half it will be there and question solving is more important but here basically what happens here is questions con I can integration in just 15 minutes but question you should learn that first let's go to first question especially chapter what I saw is that there's kind some kind of rep and recently questioned question from this chapters okay even recent but of course always one question will be the tough if they give four question one question okay next a a very very simple very very simple so this is actually the answer becomes modulus a square modulus b square and that will become 144 and they given already so they given mod a as this one so mod a is basically mod is I think maybe you're not understood will a CR B modulus a modulus B and then then sin Theta square and mod if I take common mod a s mod b sare cosare so this becomes one so this becomes what modulus a square into modulus b square so basically squ so this square plus this Square should be equal to modulus a square plus modulus B squ standard because this question has asked so many times I don't know why a get so based on this so they have given this full thing as 144 144 so we know that this is equal to this one mean this is equal to 144 and they given modulus is 4 4 4 are 16 this becomes 16 so 16 I bring it this side so now I want modulus b square that becomes 144 / 16 and I want modulus B that is root of 144 12 root of 6 is 4 4 3 are 12 answer is three it's very simple question mind this question was asked in KCT 2023 of course question actually in fact there's a very good question one question in k23 is very nice and next suppose let's take one more question similar type I want you to get the answer without I always tell you should never put your pen on the paper and out of 60 Questions 30 35 questions you don't have to put Maxim I'm not saying every question going to be easy there are some questions which even the teachers who said the paper can't solve it in one minute I can challenge at least five questions I can tell you this even you go go take the C paper and go to give a teacher and ask them to solve 60 Questions 60 Minutes 100 % I'll write and tell you even if they J Mains teacher Advan teacher also they can't solve 60 Questions in 60 minutes in C of course you'll get some extra minutes also but there are some question which take more than 3 to four minutes so C is not about C is not about you scoring 6 of 60 60 out of 60 you even the last year last year math Stoppers tho those got Les less than th000 in mathematics or the Marxist G 45 46 46 something 44 usually people score more in the chemistry because the theory Theory subject lowest in physics second in maths and highest in chemistry usually if you see any any anyone's marks in the C there will be some questions in C which are very high this question next A cross B it is very very simple A cross B square is a square actually so that's nothing but same question go modulus a square modulus B sare = 144 B 6 is what is for this two very simple you know the P it's very very important mean that next again look at this question you should tell the answer what's answer for this we already know the what is the answer become for this modus a square modus b square Ro so we get mod a into modus B that means the answer is simply 16 into 4 that is basically 64 answer is 64 64 answer is 64 basically 16 into 4 is 64 that's very simple question answer for this is 64 let's go for next question now next again see this question as in 2023 2023 question a plus b vectoral Aus B Vector if vector vector equal what is the answer for this actually are there coincident incline to are 60° see I don't the 60° I don't know how it is because they not given any values there something like 30° I mean something like root3 something is given means we can think about 60 nothing is given and a and b are perpendicular A and B are parallel so what we should do is these kind of questions very simple first you canally solve this when you can have such a way that a plus b should be equal to a minus B magnitude so I have to take the B Vector in this direction so that I take a b Vector like this the magnitude will be this is a magnitude so now I take negative of B Vector this minus B Vector still you get magnitude same only but opposite of course direction will change but magnitude will remain same so this is Possible only when if the vectors are perpendicular if the vectors are par how how is par how it's how it's possible if they parall if the two vectors are parall a plus b Vector a a vector minus B Vector obviously if there two are how even that's not possible simply you can say there perpendicular answer isend for this meod can do method also very I can tell you go to so many questions are there where a plus b a minus B is a very common question asked in the C A Plus don't Form B sare 2 a a b square A square B square- 2 a 2 Aus 2 a a square a square cancel b square B Square cancel 2 so 4 must Beal Z 4 a that means a b must be z a b is z only when they're perpendicular Z only when perpendicular I'll show it but just watch you have to because many times I can show some questions okay get this is equal to this Square equal this is equal to this so this becomes now what what happens modulus a square plus modulus b square modulus b² minus 2 a a do B and this this cancel and this this cancel and this will bring this side so we'll get 4 a do B should be equal to Z when is do product is zero that's possible only when they when they're perpendicular to each other the angle is answer is 90° answer is 90 they perpendicular to each other next 2022 Alp equal IUS 3 J Plus b equal to I + 2 Jus K cap then Express B in the form of Bal B1 + B2 where B1 is parall to Alpha and B2 is perpendicular to Alpha and B1 is given by so they're asking so this is a question they given what is the question they given see sometimes when you read don't do the question go for next question because there can be easy question with last last the first thing is you have to go through all the questions sometime you read a question [Music] where some question question don't wait see it is all about timings timings is very important anyway these questions let's see whenever they see whenever this question bet bet okay1 is basically parall to B Alpha that means B1 is Lambda time of alpha B2 is perpendicular that means what the B2 is perpendicular whenever they say perpendicular to Alpha that means the always dot product is zero so when I say B2 perpendicular to Alpha that means what you can see the B2 dot Alpha should be zero this is two conditions try to go to options clearly saying is B1 is par to Al L it can be one into this 2 into this into this exactly you may multiply with two for example 2 - 6 is both answer or maybe you may multiply this withus 3us 3 and plus they all parall to this because ratio 2 1's 2 minus 3 - 3 1's if you want - 3 1's - 3 again- 3 1 is to- 3 1 is-3 ratio of the same so these two options cannot be the answer because both are plus and plus so we got now two two more options in this case so Che 1us 3 1- good well and good you can have anything Lambda 5 by 2 doesn't matter exactly same so somewhere I feel I think I don't think so it should be same question because they clearly mentioned B should be B1 plus B2 if already if a condition go I don't think so it become same answer I think in fact this question C they had given Grace marks because I think if you solve the question you will not get any of the answers because you know mistake I think it should be supposed to be uh you value I think some some going to come I just let's calculate first Lambda this question because we can do B2 is perpendicular to Alpha but what they given is B1 and B2 findus so if you want to find B2 I should do basically B B minus B1 B B1 B minus B1 B minus B1 what is B my B is i21 so B minus B1 and know 1 - Lambda that is 1 - Lambda B minus B1 1- Lambda I Lambda * 2 so next again you have it minus B1 so what is my B1 is lamb okay Lambda so Lambda of Al alus Lambda time of so you have to multiply Alpha B and know 2 so 2 minus Lambda * of that minus of 3 3 Lambda + J cap and that minus this is K cap I'll tell how it has come first of all will be some there will be some questions in C which will definitely take more than one minute which you should not focus because there will be so many questions in C which can be solved faster than that there's so many questions in C which can be solved very fast but there will be some questions in C which will take your time so they they want to test you how much time you will you will put on this for the concept Clarity but go for next question in this case let's do what is it this minus this what is your B1 is Lambda I'll just write all the steps Lambda times of alpha Lambda I minus 3 Lambda JC so basically now we are B to find one so 1us Lambda I next you have plus so next this minus what is this beta that is 2 minus ofus 3 that is 3 Lambda plus plus 3 lamb J bet that means betus Z that is minus K only because there is nothing is there this dot product should be zero Andre we have to find dot product with again Alpha that is basically 1 and minus 3 so 1 and- 3 1 into this 1 into 1 - Lambda and that will - 3 into 2 + 3 Lambda that should be equal to Z 0 that means 1 - Lambda minus uh 6 - 6 - 9 Lambda = 0 1 - 6 is 5 = to - 5 so - 5 = Lambda 10 Lambda so Lambda is equal to basically - half lamb isalus half so my Beta 2 will be beta 1 will be - half * the alpha what is it I cap minus 3 J cap I think they made a mistake in CD I think they should have given option here maybe the answer should have given like minus half into this then this would be the answer but C mistake question not g correct answer so they will give gra marks for everyone doesn't matter don't worry on that so you have to some mark because there's no negative marking you have to click you have to take every every question clear again so this is a very lengthy question so you have to find First Alpha then perpendicular condition then you know P this is very small there so many question where they give perpendicularity and ask you to find the Lambda value this is very common question C so you to take the Lambda times and get the answer like this okay anyway let's let's come to next question okay see this question has come in KC 2018 know a that means 1 into mu that is Mu Lambda into 1 that is Lambda that's 2 into -1 that is -2 that should be equal Z so simply I can say mu + Lambda or I can say mu + Lambda asking to find Lambda and mu right just check in which case Lambda plus mu is 2 7 + 1 8 8 by 4 2 perfect 7+ perfect okay 4+ this going be the answer 9 + 1 10 10 / 4 will be not equal to 2 what about this - 1 + 9 - 9 + 8 8id 4 that is two o also two that means this answer cannot be the option any of these three can be the answers next mod AAL mod b sometimes you two options but very sad only one got eliminated this isal modulus of that that is 1 + Lambda s + 4 1 + 1 + Lambda s + 4 should be equal to Mu ² + 1 - oneus one Lambda s so I can say Lambda Square Lambda square + 5 = to Mu s simply I'm writing steps not required again so Lambda Square so I can say so Lambda Square - mu^ s is equal to -3 that means what Lambda Square s- 3 mean that means Lambda is smaller than mu and Lambda Square mu Square isus 3 Al so Lambda should be less than so Lambda first value Lambda should be less than correct oh this cannot be because this Lambda is this is more than this cannot be the answer option so this case is not getting eliminated after two steps also okay no problem okay check final step check Lambda sare Lambda Square this Square minus this Square should beus 3 1 by 16 7 7 49 49 by 16 exactly 48 by 16 the answer is three very simple so correct perfect bir so this is satisfying all the condition I don't think anything else will satisfy because this also not get satisfied check not going to satisfy this is the easiest way to do this question option first of all when you see two conditions are given when you get one condition try to eliminate the option with only one condition see the answers basically but sometime some questions will not get eliminated again check the condition which satisfies elimate it but if you want to very simple simp Lambda = Lambda so get Lambda Lambda 2 in the place of Lambda put 2 minus mu then Square it you'll get a quadratic equation solve the quadratic equation then you going to get the answer that's very lengthy next let's go for next question next if un vectors is angle between the A and B then that Sin Sin Theta is what first of all when you read a question you not get to know what to do because they given a and b are unit vectors is angle between then find the imagine I'm telling you just read the question read the question with without the options you will not get to know what to do don't know what to dotion options a plus a plus b Aus B A oh I need to do a plus b and Aus B and I told whenever a plus b a b is in vectors always Square it off Square you'll get because they given modulus a modulus a b values because so they given modulus a modulus a is unit Vector modulus B is unit Vector that mean this is one a plus b square A square B Square 2 A Plus means you'll get like a square + a square 1 square + 1 square 2 into 1 1 cos I'll get like something plus cos Theta when something plus cos Theta comes we have 2 2 coset half angle formula 1- cos 2 Theta 2 sinare that means I should not Square the Plus Things the same question was cos then you supposed to find these plus values plus I can show you why it is B the whole Square modulus a sare plus modulus B sare plus 2 into a vector do B Vector basically they given this is one this is one because 1 square + 1 square unit vectors 2 into mod a modul b cos Theta that means 1 into 1 into cos Theta so if I take this is 2 this is 2 cos Theta 2 + 2 cos Theta so if I take two out I'll get 1 plus cos Theta is there any way to get sign from this I mean you can maybe do something and get it very lengthy but I can directly do minus no instead of Plus instead of plus here instead of plus simple 2 coset 1- cos 2 Theta sin Square Theta by 2 that is 2 and that is 2 * sorry 1 - cos is 2 * of sin sare Theta by 2 so 1 - cos Theta 1 - cos 2 Theta is 2 sin sare thet very very important integration differentiation it's very easy 1 - cos Theta is basically 2 sin s Theta I need to find this I need to find what they given is oh you need to get in terms of this only right okay so modulus a and modulus B the whole Square 2 into 2 4 C you should not write any steps it is not board exam that is very important next is 2 mod b is 3 and then angle between A and B is 120 when 120 is given suppose Square don't try to always this is the wrong way of doing it see a s + b s- 2 a B positive from 0 to 90° so cost 120 must be negative value that it's very simple 180- 60 60 isus cos 60 that means- half 180 9 second qu cos is negative pi minus cos Pius Theta is cos minus cos Theta this is very important iners trigonometry trigonometry in fact when you want to find the negative vales for example inverse of xus cos invers x sorry cos inverse of cos inverse ofx equal to P minus cos inverse of x invers t you know so basically anywh so this value is birth actually 2 into modulus a by 2 again 2 by 2 is 1 again 3x 3 is 1 again you know you have minus half here in this case simple so 2 into 1 into minus plus 2 2 cancel 1 + 1 2 2 + 1 3 answer is three simple simple it's very simple you able to find these values very fast if you're still struggling to find this value is the first chapter you should study trigonometry the first chapter first formulas you should study complete trigonometry form you can Sol other questions this complete should be values 2 answer isar again you have to expand this model A S Square by 4 + modulus b² by 9 in the not required because a is two they already given so 2 square + 2 square + 4 4 anywhere 2 by 2 is 1 no but 2 square and the 4 4 by 4 1 one that's why I return here minus modulus a by 2 modulus B by 3 and it's basically minus off is not required all the things again this is also one this is also one two two get cancel plus one 1 one one 1 1+ 1 + 1+ answer is next the area of a param again this question is repeated two or three times in C the area of the param is with A and B is as two adjacent sides of 15 square units then the area of the parallelogram having 3 a vector plus 2 B vector and a vector VOR has two ad sides of okay a vector B Vector instead of A and B I'm taking other two vectors that is 3 a vector v good very simple you just if you want to find what you want do you do cross find cross that means what so 3 VOR a cross product a cross a means you're doing a cross product with the same Vector a cross a b CR c c is always 3 a into a that becomes zero 3 a into 3 B that becomes next 2 B into a you going to get it now plus 2 B Vector into a vector I'm doing B into a b Vector into a again 2 B into 3B answer is 6B get cross product B it's not commutative A cross B is negative of B Cross C that means 9 - 2 that is 7 that means 7 * of what is the answer 15 so 7 * of 15 so 7 * of is whate a CR B minus two * of a someone is struggling to understand this again don't worry go for next just go for next question if you not understood I'll explain one again see M Vector 2 vector a vector plus this into this a into a that is Zer basically because so basically B not get answer A and B A into a CR B Because B cross a is negative of A cross B then already 9 - 2 is 7 so you get 7 A cross B so A cross B is already given this value is already given as what 15 so 7 15 times that that is one that's very simple question I think there are two more question like this similar question avoid I have not taken all all the question of C you check one the question what Tak is different I mean first two first two question because that question has come so many times next what's for this if angle between A and B is minus between A and B is 2 pi by 3 2 pi by 3 and 120 again 120 120 120 120 is basically minus half next okay prodction of a vector on B Vector on the on the direction B is minus 2 on the a vector on B Vector a vector on B Vector simply we can say modulus B cos a vector on B Vector sorry uh modulus sorry a vector mod a COS Theta that should be equal to minus 2 cos Theta isus halfus minus cancel so a vector so a vector is equal to half shift four the answer is four question C this question is come K 2019 very simple questions I hope you understood a vector on B Vector modulus a b a do b/ by modul of B that's nothing but you'll get modulus a COS Theta because this will become modulus a modulus B cos Theta modulus B mod cancel you get a COS Theta they given this value asus2 so and the COS Theta they already given angle as minus half they have not given C they told they telling that 120° cos 120 is cos 180 - 60° that isus cos 60 minus half if answer is four next let's go for question unit Vector is perpendicular you have to check for perpendicularity condition the plane contain the vectors oh this Vector there is a vector which is perpendicular to the plane containing this vector and this Vector know Vector simple this is common questions in C sorry but answer answer this question just in one minute very simple so they given answers don't think think that now we can check for perpendicularity right that means what is given should be perpendicular to both of these I'll check any one first I'll check with this some should be minus so this is not possible next okay plus not possible minus not possible there should be some plus some minus so that we get zero if you multiply this 1 into for example 1 into- one 2 into -2 1 into- all get like -1 - 2 -1 they not get add they will not get know 1 into 1 1 2 - 1 so this not zero 1 sorry -1 -1 2 -1 yeah this is zero so option is B actually not I'll teach once again for next one what I'm say what I'm saying is coefficient 1 2 1 1 1 1 1 1- one the direction ratios forget about ro3 ro3 is there -1 1 and - one and -1 -1 -1 what I need is If I multiply this one to one with these values I should be able to get zero for perpendicularity in this case there's no question at all because all are positive 1 1 2 1 all positive in this case you can check 1 into 1 2 3 - one again not not possible what about here it's possible 1 into -1 is -1 2 into 1 is 2 is therefore Z therefore answer is this i j values 1 2 1 and - 2 1 next two vectors I plus J + 3 and this one represent two sides of triangle ABC the length of the median median Medi a vector B Vector 1+ 1 1 + 1 one 1 1 1 + 1 by 2 that is 1 that 3 + 1 by 2 that is 2 5 + 1 by 2 that is 3 the root square square square root of this if you want to find the length magnitude so the answer will be 9 10 14 answer isun 14 clear again 1 one 1 1 3 5 you should never write a single step every single every single letter you write in C will consume 1 + 1 2 by 2 is 1 1 + 3 by 2 2 1 what you divid 1 + 1 sare + 2 square + 3 Square magude find magnitude that means root of a square + B sare + C Square I mean the this answer is root 14 next if a vector and B Vector mutually perpendicular unit vectors already perpendicular that means what if that modulus if there a vector I think modulus should not come they vectors not modulus vectors vectors okay so a vector and B Vector mutually perpendicular that mean answer is z a a into a is always you get that value you get some value but cross product a cross a zero 2 into 6 12- 12 answer is three answer is three for us are clear again explain this 15 modulus a modulus a so because when you have a into a is one basically it is what happens vectors unit vectors ular perpendicular that means what you get zero again B and are perpendicular A and B means b and a do product is basically 12 that is B into B again B is one again B is one and angle between them is 0 that is C 0 that's again one so you'll get basically 15 - 12 the answer is three next let's go for next question next next direction cosin of an Vector dire asking what is the vector negative of that you can have minus plus minus also plusus plus this can be possible all plus can't be possible all plus can't be possible plus minus plus this is possible okay but ratio should be same always Direction 2 2 2 answer is this simply given direction to mulp answer with three then you're going to get the vectors that is 2 I cap minus J cap plus 2 K that's it because it magnitude is the when you multiply unit Vector with the magnitude you're going to get the vector actual Vector clear again next so for unit Vector Direction cosin and Direction ratios both are same for unit Vector next a is 3 B is 4 C is 5 each one of this A and B is are perpendicular to sum of the remaining two actually what they saying that a is perpendicular to sum of the remaining two that means B + C B is perpendicular to a + C C is perpendicular to a + b this is what said that's the meaning that means perpendicular means Z 0 Z what is a c C Square 2 a 2 BC 2 C you learn this in where in your nth class a square + b square + C square + 2 a b + 2 b c you get like a b again plus b do c so if you keep adding all these things right you will get totally 2 ab 2 BC 2 CA all should be equal to zero instead of saying this should be zero they given indirectly that should be that mean this should be equal Z and they're asking what is the value of this they given the individual values that is 9 + 16 + 25 25 + 25 that is 50 that is this whole square is 50 if you w want square that is root of 50 so root of 50 is 25 into 2 25 into 2 root 25 is 5 and then ot2 cunk 50k otk 25 into 2un 25 is 5 and then that is < tk2 5 < tk2 answer is what 5 < tk2 next cos beta cos G the direction COS of the vectors cos 2 you can write cos 2 2 Alp in terms of sin alpha or cos Alpha you can write whatever you want but I in terms of because they given that means they told something like that okay 2 cos + 1 oh 2 cos Alpha + 1 basically what's the value of cos 2 Alpha so cos 2 2 cos² Alp + 1 okay next just calate all the values 2 cos² Al + 1 so plus 2 cos² 2 cos Square beta + 1 cos 2 GMA under 2 cos² gamma + 1 so you can take two common I'll get cos Square cos Square beta you'll get like cos Square Alpha + cos Square beta + cos Square gamma + 1 + 1 + 1 + three what's the formula C is what formula is 2 cos² Theta - 1 1 - 2 sin okay so minus minus I was wondering option actually actually in case in this question if there given answer five n c f very common teachers also do mistake cost is 2 cos - 1 it's- one here it's basically minus one -1 -1 -1 basically minus 3us 2 - 3 is- one answer isus oneus one Ang formas I mean two many are told actually cos is 2 cos minus one yes correct cos sare 1 cos 2 Al 2 cos² Alpha - 1 2 cos² Beta - 1 2 cos² gamma - 1 so 2 2 2 you take two out you get cos Square Alpha + cos Square beta + cos Square gamma -1 -1 - 1 that is - 3 = Z so I'm writing in pink color girls favorite so if this formula come they should not forget uh 1 - 3 = 0 2 - 3 that's equal to -1 answer is-1 next let's go for next question now C ask 2017 a plus b plus C the whole square a plus b plus c sare a s + b square + C square + 2 a b + 2 BC + 2 CA and they asking what is the value of this that means and they're all unit vectors so that means a vector plus b Vector plus C Vector don't write vectors and all just write a plus b plus C is fine okay so square is equal to a square that is 1 b square + 1 plus C square is also 1 2 * of they're asking what is this only okay this value oh they given this is zero that means what this is three if this side minus 3 = to 2 into something asking what is this only that is basically minus 3 pu textbook the ncrt question clear again okay so that's all this is even the remaining P go and check it mostly you can do it but there will be don't have on that I hope you understood if you're interested you can also join simplified courses you can join any of the courses and that's it see in the next videos bye-bye take care

Channel: SimplifiedMinds KCET

Views: 39,500

Keywords: kcet maths, kcet vectors, kcet maths oneshot, kcet pyqs, kcet simplifiedminds, kcet vectors important, kcet maths simplifiedminds

Id: ZtUE1MC265g

Length: 66min 37sec (3997 seconds)

Published: Mon Apr 08 2024