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Direction of vectors from components: 1st & 2nd quadrants

Video transcript
so we're gonna do in this video is look at a series of vectors and we're gonna draw them in standard form where their initial points for their tails are gonna sit at the origin and receiver can figure out the angles that they formed the positive angles that the form with the positive x-axis and like always positive influence if you could figure out what these things are going to be on your own gonna figure out in degrees so in this first one I have the vector you if we were to write it in a unit vector notation sometimes is called engineering notation with its three times the unit vector in the horizontal direction I unit vector plus four times the innovator in the vertical direction or you can view this as the x component is three and the y component to score you see that here if you start at the origin we're going to move three in the horizontal direction and we're gonna move for in the vertical direction now to figure out this data there's a couple of ways to think about it we could just construct a right triangle for this one in particular we dedicate the x coordinate is three so if we were to create a right triangle here this side would have a length of three and then we have a height of for the y coordinate is for so this side has a length of four and we know just from even our basic so Couture definitions of trig functions that what trig function involves the opposite of an angle so the opposite of the angle and the adjacent of an angle wilt engine does so katoa so we know we know that the tangent data is going to be equal to the length of the opposite side which is for over the length of the adjacent side over three and so we wanted to solve for theta we could just say that they do is equal to the inverse tangent sometimes people say arc tangent of for thirds of war over or over three and let's evaluate this and I'm gonna get my calculator out to do it so I wanna take 4/3 which is that and how to get press this button right over here that takes my tangent to mix into the inverse change it so I will take the inverse tangent of this and i get i get roughly fifty three point one degrees this is approximately fifty three point one degrees which looks about right this looks like a little bit more even though I didn't completely sober precisely I had drew this looks a little bit more than a 45 degree angle so that that feels that feels pretty good other than you might have said they will look for is the y component three is the x component and so maybe tangent of data is always going to be the y component over the excavated and that in fact is the case and that actually comes straight out of the unit circle definition of the trig functions where you could say that if you have a unit circle if you have a kid right over here so let me draw the coordinates so if I Drive unit circle right over here and if I were to have some line here thinking about vectors that the angle for the positive x-axis the tangent of that angle tangent of the day is going to be the y coordinate where we intersect the circle over the x coordinate and so you can imagine if you made a unit circle right here the unit circle right over here the ratio between the y coordinate the exported of this point right over here where we intersect the circle is going to be the same thing is the ratio between four and three so this is going to be this also is going to be wanted one third or fourth thirds so either way you can think of when you think about vectors the tangent of the angle that forms of the positive x-axis is going to be is going to be the y component over the x component so it's nice when everything kind of fits together like that so let's levers that to figure out this to figure out this single ride over there well we could say that tangent is going to be equal to the y component the y component over the x component over a new color or see over negative fides it's going to be negative negative six minutes or we could say data so let's be a little bit careful so week we could say we could say that data is equal to is equal to the team the inverse tangent that same color in worst inverse tangent of straight like this negative 650 people question mark here to see if we feel good about it the answer would get one we do this so let's do that so if we do if we do 6/5 would you go to that and we want to make this negative so that's a negative six minutes and we take the interests and it's already press this button so there's going to be a nurse tended not to mention I get negative roughly negative fifty-point two degrees so this is approximately negative fifty point to agree what does that look right or not they looks like it's over ninety degrees negative 50.2 degrees negative 50.2 degrees is actually giving us is actually giving us this angle right over here so that's giving us that specifying another victory about a line and other line that would have the same handed seemed as it's really important to visualize it think about this is the reason why is because the arc tangent ore the inverse tangent functions in calculators they will give you an angle that is between negative nine degrees and positive ninety degrees so something that's in the fourth or 1st squadron well here we have something that's in the second quarter as we have to make sure that we're thinking about it right so that we can make the appropriate adjustments so that would give that would give the case if we were looking at a a line like that or vector like that so in order to figure out what in order to figure out the actual angle what we want to do is add a hundred and eighty degrees to it so to get the victory that goes in the other direction so you want to add so there is going to be equal to or I could say approximately equal to negative 52 degrees plus one hundred and eighty degrees so let's do that so let me + 180 is equal to approximately 120 9.8 degrees so they is approximately equal to 100 29483 said I get that right very short memory oh yeah that's right what are 29 48 degrees and that looks much like that looks much better that looks clearly is an angle we're going to go that takes us to ninety degrees and then we're going above 90 degrees now another way you could have thought about this is what you know from from Sokha toy and just right triangles we could construct a right triangle where we construct a right triangle using some colors well we know so I can just drive like this so this is it and what is that height well we started is going to be six and then the base the base rate over here what is that like the going to be well we know that we're going from the origin remain five bags if we think I just the absolute value the length of that line that is going to be five and so we could figure out is using right triangles this single be recalled that acts and so we could say the tangent of Acts tangent of AX is equal to six over 5 opposite over adjacent six over 5 war that X or you could say access you go to the inverse inverse tangent 6 this and what is that going to be so if we take 6/5 1.2 and then we're going to take the inverse tan that we get approximately fifty fifty point two degrees so acts is approximately fifty point two degrees which looks right but remember we're not trying to figure out acts were trying to figure out we are trying to figure out what they did what they did is and you could see that accident data are supplementary so they'd up plus access going to be a hundred 80 or thing is going to be a hundred 80 minus that 280 minus that just put a negative sign in front of that and add 180 and that might look familiar so that gets us to 129 roughly 100 29.8%