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Current time:0:00Total duration:9:46

Calculating dot and cross products with unit vector notation

Video transcript

so far when I've told you about the dot and the cross products I give you the definition as the magnitudes times either the cosine of the sine of the angle between them but what if you're not given the vectors visually and what if you're not given the angle between them how do you calculate the dot in the cross products so let's say well let me give you the definition that I've given you already so let's say I have you know a dot B dot product that's the magnitude of a time's the magnitude of B times cosine of the angle between them a cross B is equal to the magnitude of a time's the magnitude of B times sine of the angle between them so kind of the perpendicular projections of them times the normal vector that's perpendicular to both on the normal unit vector and you figure out whether it's either you know which of the two perpendicular vector it is by using the right hand rule but what if we don't have them visually define and we don't have the Thetas the the angles between them what if for example I were to tell you that the vector a the vector a what if I were to give it to you in engineering notation an engineering notation you're essentially just breaking down the vector into its XY and z components so let's say unit vector a is 5i I is just the unit vector in the X Direction minus 6j plus 3k these are all IJ and K are just the unit vectors in the x y&z directions and the end of five is how much it goes in the X direction the minus six is on which goes in the Y direction and the three is how much it goes in the Z direction you could try to graph it and actually I'm trying to look for a graphing calculator that will that will do this so I can show you all in videos to give you more intuition but let's say this is all you're given and let's say that B B I'm just making these numbers up let's say it's minus 2i and of course we're working in three dimensions right now plus I don't know seven J Plus 4k and you could graph them but obviously if you were given a problem and actually if you if you're actually trying to model vectors in a computer simulation this is the way you would do it you'd break it up into the XY and z components because in to add vectors you just have to add the respective components but how do you multiply them either taking the cross or the dot product well it actually turns out I'm not going to prove it here but I'll just show you how to do it then the dot product is very easy when you have it given in this notation actually another way of writing this notation sometimes in bracket notation sometimes they would rewrite this as 5 - 6 3 where's just the magnitudes in the x y&z direction just want to make sure you're you're you're comfortable with all of these various notations you could frighten written B as minus 2 7 4 these are all the same things shouldn't get daunted if you see one or the other but anyway so how do I take a a dot B this I think you'll find fairly pleasant all you do is you multiply the I components add that to the J components multiplied and then add that to the K components multiplied together so it'd be 5 times minus 2 so 5 times minus 2 plus minus 6 times 7 minus 6 times 7 plus 3 times 4 plus 3 times 4 so it equals what is this minus 10 minus 42 plus 12 so this is minus 52 plus 12 so it equals minus 40 that's it it's just a number and I'd actually be curious to graph this on a three-dimensional some type of graph er to see why it's minus 40 they must be kind of going in opposite directions and their projections onto each other go into opposite directions that's why we get a minus number and then we you know well I you this the purpose of this video I want to get too much into the intuition this is just how to calculate but it's fairly straightforward you just multiply the X components add that to the Y components multiplied and add that to the Z components multiplied so whenever I am given something in engineering bracket notation and I have to find the dot product it's very almost soothing and and and not not so error-prone but as you will see taking the cross product of these two vectors when given in this notation isn't so straightforward and I want to keep in mind you know another way you could invent you could have figured out the magnitude of each of these vectors and then you could have used some fancy trigonometry to figure out the Thetas and then use this definition but I think you appreciate the fact that this is a much simpler way of doing it so the dot product is a lot of fun now let's see if we could take the cross product and I'm not going to once again I'm not going to prove it I'm just going to show you how to do it in a future video um I'm sure I'll get a request to do it eventually and I'll prove it but the cross product this is more involved and I never look forward to taking the cross product of two vectors in engineering notation a cross B it equals so this is an application of matrices so what you do is you take the determinant I'll draw a big determinant line on the top line of the determinant and this is really just a way to make you memorize how to do it it doesn't give you much intuition but the intuition is given by the actual definition if you know how much of the vectors are perpendicular each other multiply those magnitudes right-hand rule figures out what direction you're pointing in but the way to do it if you're given the engineering notation you write the ijk unit vectors in the top row i j k then you write the first vector in the cross product cuz order matters so it's five minus six three then you take the second vector which is b which is minus two seven four so you take the determinant of this three by three matrix and how do i do that well this is equal to that's equal to the sub determinant for i so the sub determine for i if you get rid of this column in this row the determinant that's left over so that's minus six three seven four tie i I want to review two determinants if you don't remember how to do this but maybe me working through it will just will will jog your memory and then remember it's plus minus plus so then minus the sub determinant for J what's the sub determinant for J well you cross out J's row and column so you have five 3-2 four so five 3-2 four we just crossed out J's row and column and was our four left over those are the numbers and it's sub determinant that's what I call it J plus I always I want to do them all in one line because it'd have been a little bit neater plus the sub determinant for K cross out the row and the column for K if - five - six five - six - 2 + 7 - 2 and 7 times K and now let's calculate them and let me make some space because I've written this too big I don't think we need this anymore so what do we get let's do again let's take this up here so these 2 by 2 determinants are pretty easy this is minus 6 times 4 minus 7 times 3 so it's - I always make careless mistakes here minus 24 minus 21 times I minus 5 times 4 is 20 - minus 2 times 3 so minus minus 6j it's going to do this in the next row plus 5 times 7 35 minus minus 2 times minus 6 so it's minus positive 12 K could simplify this which equals minus 24 minus 21 it is minus 35 I didn't have to put a parenthesis I and then what's 20 - minus 6 well that's 20 plus plus 6 or 26 but then we have a minus out here so minus 26 and those 20 35 minus 12 s 23 plus 23 okay so that's the cross product and if you were to graph this in three dimensions you will see and this is what's interesting you will see that this vector if my math is correct - 35 I minus 26 J + 23 K is perpendicular to both of these vectors anyway I think I'll leave you there for for now and I will see you in the next video and hopefully I can track down a vector graphing program because I think it'll be fun - both calculate the dot and the cross products using this met the methods I just showed you and then to graph them and to show that it really does work that this vector really is the right hand it really is perpendicular to both of these and pointing in the direction as you would you would predict using the right hand rule anyway I will see you in the next video