Column Vectors – Corbettmaths

column vector
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in this video were going to look at column vectors a vector is something with a direction and a magnitude or size and the column vector is one of the ways to represent in these the number on the top represents how far to the right or to the left Im if its positive to the right if its negative to the left the number of beneath represents up were done so if its positive upwards and if its negative downwards you may have encountered them in a topical translations also so for instance if they said translate ship a by the vector 3-1 youd move it three squares to the right on one square upwards okay now what were going to do is were just gonna represent these visually okay were just gonna draw okay so this vector is 3 1 which means three to the right on one up so one good is Im just gonna choose any starting position and Im gonna go 3 to the right Im one up and then that would be that vector Im just going to draw an arrow on it just to show were going this way okay were going free to the right and one up when were not going through the left and one down and thats it so Im just gonna represent these column vectors visually just for the moment okay heres another column vector this column vector is minus 4 2 which means 4 squares to the left and 2 squares up so if I wanted to represent that visually what Im gonna do is Im just gonna go four squares to the left one two three four two squares up and again Im just gonna make sure I put my arrow on it like so just to show that were going to the left and up next one one to the left or the very the column vector is minus one minus three so
The Voltreport column-vector-0 Column Vectors - Corbettmaths
its gonna be one to the left three done so against choose a starting position because Im gonna go three down Im gonna choose starting position over here somewhere and its one to the left Im three done one two three and again just put an arrow on it just to show were going this way to the left and done finally weve got the column vector at 0 minus 2 that would mean 0 squares to the left or right and to square silence is just going to be straight down to squares so if I choose starting position go down to and again just represent with the letter O okay so thats how you represent column vectors graphically or visually now what Im going to do is Im going to do some addition or subtraction using column vectors so Im here weve got the vector a now as you can see the a is in bold whatever ran these on the page usually I would do an underline it but whenever theyre written in a textbook on a computer or in an example sometimes they just sort of have a printed and bold its a bit hard to write in bold so thats why I would write nuns line okay but weve got the vector a which is one two that means one square to the right and two squares upper and weve got the vector B which is 2 minus 1 which is two squares to the right on one square done weve been asked to add a and B together now what Im gonna do is Im just gonna draw up a and then Im gonna draw B so a is 1 square to the right and two squares up so thats the vector a Im just going to put an arrow on it and Im just gonna write a beside it now B is two squares to the right on one square down
The Voltreport column-vector-1 Column Vectors - Corbettmaths
so where I finished off with air which was here Im gonna go two squares to the right on one squared done and again Im just gonna put in the arrow on it and call that B now as you can see Ive gone to square one square to the right and two up and two squares the round one down and overall I finished him now if I was the done one one journey I could just go straight from my starting point to the finishing point here and altogether that would be three squares to the right one square up so that would be the vector if Im going 3 to the right on one up that would be the vector 3 1 so Im just gonna bring them up here and got add them together I have 1 plus 2 is 3 and 2 plus minus 1 remember when you add minus 1 it goes back down to 2 plus minus 1 would be 1 as you can see thats the same as the journey just go from the starting point to the finishing point 3 to the right 1 1 up so to add column vectors all you do is add the top numbers and add the bottom numbers and thats it ok so lets have a look at some questions so were going to do these two about drawing diagrams so weve got the vector air which is six one and weve got the vector B which is minus four or five so if I was asked to find the vector of air plus B thats what Id get that out of them together so six plus minus four well six plus minus 4 would be the same as six tick or f4 well its gonna be two and then 1 plus 5 1 plus 5 is 6 so if I was to go six squares to the right on one square up
The Voltreport column-vector-10 Column Vectors - Corbettmaths
and then four squares to the left and five squares up all together that would be two squares to the right and six squares up Im and thats it and thats how you dont used to add them together okay this time were gonna multiply a vector by a scalar or scalars just a number okay so Im just gonna multiply or double the vector a so the vector a is 6 1 well if instead of course 6 that the right 1 1 up if I was to double that that would be 12 to the right and 2 up so if youve got a number in front of a vector you just multiply the numbers by whatever numbers in front of the letter so because it was a 2 I double booked for these numbers if it was a 3 I would multiply the book by 3 and so on okay our next question an expiration weve been asked to find 3a plus 2b so what Im actually gonna do is Im gonna write down what free a is and Im also going to write down what to be is and then Im gonna add them together afterwards I do like to show my work and I whenever Im in the exam it just it also it just helps me work it out so free air well three times they would be eating and three to be or doubling this would be minus in and 10 and then adding them together well 18 plus minus see it well 18 plus minus here its gonna be 10 because remember when you add a negative number of goes down so its got to go down by here and then Im gonna do three add 10 and 3 add 10 is 13 so 3a plus 2b would be 10 for a tape next now were going to do subtractions whenever youre taking them away again you just take them
The Voltreport column-vector-11 Column Vectors - Corbettmaths
away its just important to remember which one you know you start more as a minus B that will be a different answer to be take were okay so B minus a minus B so a minus B thats 6 minus my four was gonna be ten because youre subtracting a negative and one minus five from 1-5 will be minus fourth next be minus a or b tick away so youve got minus 4 take away 6 thats minus 10 and then weve got a 5 take away 1 & 5 tick where 1 is 4 and lastly weve got minus 5b so what were going to be doing is were going to be multiplying B by minus 5 so the vector B is minus 4 5 were timesing the book by minus 5 so minus 4 times minus 5 is 20 and 5 times minus 5 is minus 25 and thats it lets have a look at one last example so this time weve got a is the vector minus 1 0 and B is the vector minus 4/9 and weve been asked to find 2m minus 4b so what Im gonna do is Im gonna work out with two As Im gonna work out with four beers and then what Im gonna do is take them away from each other so 2a 2a is gonna be double this thats gonna be minus 2 0 and 4 be tending these books by 4 will be -16 36 because 4 times those numbers are 4 times minus 4 is minus 16 and 4 times 9 is 36 now one has to do is to do to a takeaway for B so its gonna be 2a so its gonna be minus 2 minus minus 16 or minus 2 minus minus 16 what means youre gonna add 16 on so Ill be 14 and then 0 takeaway 36 and zero tick where 36 will be minus 36 and thats it
The Voltreport column-vector-12 Column Vectors - Corbettmaths
tags:
corbettmaths, corbett, maths, column vectors, vector addition
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