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Current time:0:00Total duration:8:07

Adding vectors in magnitude & direction form (2 of 2)

Video transcript

in the last video we were able to figure out what a plus a plus B is when you when you view it from a component point of view and so if we were to visualize that that was that's vector a and then let me let me paste vector B here it's a little bit messy so that is vector B there and actually let me see if I can clean this up a little bit so let me let me clean this off look at the eraser tool so let me clean this up a little bit so that we can see a little bit clearer so the vector that we care about almost done cleaning the vector I'll get rid of this too that's because we just want to visualize stick to actually I shouldn't get rid of that part so ideally this right over here so the vector that we care about a plus B that is this vector that is this vector right over here we could start at the tail of a go to the head of a and then place the tail of B there and then where the head of a is that's going to be the head of a plus B so that's this vector right over here and we can think of its horizontal and vertical components its vertical component is this vector is this vector right over here and that's that vector right over there and it's horizontal component is this vector is this vector which is this one right over there so what I want to do now is figure out what is the magnitude what is let's just call this vector C so I don't have to keep writing a plus B I want to figure out what is the magnitude of vector C and I want to figure out the angle I want to figure out its direction I want to figure out this angle I want to figure out this angle right over there so let's think about it let's think about it step by step so the magnitude is actually maybe this most straightforward one and let me redraw a vector C here so vector C I could actually I'll use the space right over here so vector C looks like this a vector C looks like that and it has its horizontal component if you have its horizontal component and you have its vertical monent and you have its vertical component so the magnitude of it we just know from the Pythagorean theorem the square of the horrors of the magnitude of the horizontal component plus the square of the magnitude of the vertical component that's going to be equal to the square of the magnitude of the vector or another way of thinking about it the magnitude of the vector is going to be is going to be equal to the square root of this business squared plus this business squared and that's going to be actually I'll just write it just so that we well it's getting messy so let me just get the calculator out to get a reasonable approximation so get the calculator out so it's going to be the square root it's going to be a little bit of a hairy statement this is going to be the square root of three times the square root of three over two over two so that's that minus minus square root of two minus the square root of two this is going to be that quantity that's what we have an orange right over there we're going to square that plus three over two three divided by two plus the square root of two plus the square root of two so that quantity squared and we're of course taking the square root of all of that so that's going to be equal to we deserve a little bit of a drum roll now 3.14 oh it's it felt like it might have been close to PI but it's not pi would be 3.1415 and i and keep going with this three point one four five I'll just write this approximately 3.14 six so let me write this this is approximately approximately 3.14 six is the magnitude here and that makes sense I've we saw a vector vector a right over here had a magnitude of three and as we can see this one looks a little bit longer than that even when we when we look at it when we look at it visually so that that actually is consistent with what we what we see here now let's think about the direction so let's call this angle let's call this angle I don't want to do in that dark color let's call this angle right over here theta so we know what do we know we know that well we know we know what the length of the opposite side is and we know what the length of the adjacent side is and actually now we know the length of the hypotenuse but let's just say we didn't even know the length of the hypotenuse if we know the opposite and the adjacent you can use tangent we know that the tangent of theta tangent of theta is going to be equal to the length of the opposite side over the length of the hypotenuse so that would be equal to this so copy and paste it's going to be equal to that over this over over this right over here so copy and paste alright so it's going to be equal to that so let me draw a little line here or we could say that theta is equal to the inverse tangent sometimes called the arctangent the inverse tangent of all of this business of all of this so let me just copy and paste copy and paste right over there and let's let's put that into our calculator and see what we get so let's see we're going to take the inverse tangent and I've already verified that I'm in degree mode in this calculator and we saw that over there that it was interpreting this correctly as 30 degrees not 30 radians so the inverse tangent of all right so once again 3/2 plus square root of 2 plus the square root of 2 divided by divided by this which is 3 times the square root of 3 divided by 2 minus the square root - and the order of operations the way the calculator will interpret this it should be correct and so close the parentheses there that closes that parentheses and then we have to close that parentheses there and we deserve another mini drumroll and we get sixty seven point well let's just say it's roughly sixty seven point eight nine degrees so theta theta let me write this here theta is approximately sixty seven point eight nine degrees that I get that right yep sixty seven point eight nine degrees and that also looks about right if you look at this angle right over here it looks you know if we eyeball it looks like a little bit more than sixty degrees so just like that we were able to figure out the magnitude of the sum and the direction now there's one thing that I think it's interesting to keep in mind notice that the magnitude the magnitude of this vector is less than the sum of the magnitudes the magnitude of vector a was three the magnitude of vector B is 2 and so 3 plus two would have been five but this has a smaller magnitude so the only way that the magnitude of a sum is going to be the same as the sum of the magnitudes is if both of these vectors are going in the exact same direction then they would be completely additive but if they're going in even slightly different directions then you're not going to have the magnitude of the sum being the same as the magnitude it's always going to be less than the magnitude of or the sum of the magnitudes and I'll in the next video I'll talk about that a little bit more depth