in this video we're going to talk about how to add vectors so a vector is a quantity that has magnitude and direction so let's say if we have a force vector that's 100 newtons directed towards the east the magnitude is the size of the force it's 100 newtons and the direction is east and let's say if we want to add it to another force that's directed east as well let's say it's 50 newtons the resultant sum of these two forces is going to give us a net force of 150. whenever you have two vectors that are parallel to each other you can simply add the numbers to get the resultant sum now let's say if we have a 200 newton vector directed east and 120 newton vector directed west what is the resultant vector and also specified the direction so this is positive 200 because it's directed towards the right this is negative 120 because it's directed towards the left if you add these two you should get a net force of positive 80. now it's going to be smaller than the original one but it's still directed east because this vector is greater than this one now as another example let's say if we have a force vector that's 60 newtons directed east and another one that's 90 newtons directed west what is the resultant force vector and what's the direction so this is positive 60 that's negative 90 so we're gonna have a net force of negative 30 newtons or you could just say 30 newtons directed west now what if we have a force that's 80 newtons directed north and another one that's 120 direct itself so 120 minus 8 is 40. so the net force is going to be 40 itself and if you recall this is north south east west now sometimes you may need to add two vectors that are not parallel or anti-parallel to each other so let's say if we have a third newton force vector directed east and a 40 newton force vector directed north what is the resultant force vector if you have two vectors that are perpendicular to each other you could find the resultant force vector by finding the left of the hypotenuse of the right triangle that is formed so therefore the resulting force vector is going to be the square root of f1 squared plus f2 squared let's call this f1 and this force f2 so it's going to be the square root of 30 squared plus 40 squared if you're familiar with the 345 triangle then this is going to be the 30 40 50 triangle so the resultant force is going to be 50 newtons now sometimes in addition to finding the magnitude of the resultant force vector which is 50 you may need to find the direction as well so you got to find the angle theta to find it you can use this formula it's the inverse tangent of the y component divided by the x component so the force in the y direction is 40 and the x direction is 30. so to find the angle it's going to be our tangent the opposite side which is the y value of 40 over the adjacent side which is the x value of 30. hopefully you're familiar with sohcahtoa sine is opposite over hypotenuse cosine is adjacent over hypotenuse and tangent is opposite relative to theta over at the adjacent side the arc tangent of 40 over 30 is 53.1 degrees so we can say that the resultant force vector has a magnitude of 50 newtons and a direction of 53.1 degrees relative to the x-axis let's try another example so let's say if we have a force vector that's 50 newtons directed west and another one that's 120 newtons directed south calculate the magnitude of the resulting force factor and find the direction so let's draw a triangle so this side is 50 and this side is going to be 120 so the resultant force vector is the hypotenuse of this triangle so notice that it's in quadrant three now if you're familiar with the 5 12 13 triangle then the hypotenuse has to be 130. so the resultant force vector is going to be the square root of 50 squared plus 120 squared and that's going to turn out to be 130 newtons so that's the magnitude of the resultant force vector now all we need to do is find the angle so first let's find a reference angle to find a reference angle use the arctangent formula take the force in the y direction divided by that in the x direction but make it positive initially this will give you an acute angle between 0 and 90 and then you could adjust it later so our tan 120 over 50 will give you a reference angle of 67.4 so here's the reference angle it's inside the triangle now what we need is the angle relative to the positive x-axis so if this is 180 then the resultant vector is 180 plus 67.4 relative to the x-axis so it's at an angle of 247.4 so therefore this is the resultant force vector which has a magnitude of 130 but an angle of 247.4 degrees so that's the answer now let's say if we have a force vector of 45 newtons directed east and we wish to add it to a force vector of 60 newtons directed south find the magnitude of the resulting force vector and also the angle so let's draw a triangle so this is 45 and this is 60. so let's find the magnitude of the resultant force vector so it's going to be the square root of 45 squared plus 60 squared which is 75 newtons so that's the magnitude now we got to find the angle but let's find a reference angle first so it's going to be arc tangent the force in the y direction divided by the force in the x direction so that's 60 over 45 which is 53.1 now that's the reference angle the first force vector is positive it's directed east the second one is directed south so therefore the resultant force vector is in quadrant four now if this angle is 53.1 to find the angle relative to the positive x-axis it's going to be 360 minus 53.1 keep in mind a full circle is 360 but you need to go back 53.1 degrees so 360 minus 53.1 will give us an angle of 306.9 so the resultant force vector has a magnitude of 75 newtons and it's directed at an angle of 306.9 degrees relative to the positive x-axis so this is the answer so if you know the resultant force vector is going to be in quadrant one then the angle is going to be the same as the reference angle and keep in mind the reference angle can be calculated by taking the inverse tangent of the force in the y direction divided by the force in the x direction and if you always make these positive you're always going to get an acute angle between 0 and 90 which is the reference angle now if you know the resultant force vector is in quadrant two then to find that angle it's going to be 180 minus the reference angle if it's in quadrant three we cover this one already it's 180 plus the reference angle and if you get an answer that's in quadrant four it's going to be 360 minus the reference angle which was the case in the last example so it's helpful to know these things or you could just see what to do visually once you graph everything now sometimes you may need to add vectors that are not parallel or perpendicular to each other so let's say if we want to add this vector which is a 100 inch directed east plus another vector let's say it has a magnitude of 150 but it's directed at 30 degrees above the x-axis how can we find the resultant vector first let's re-list what we have the first vector has a force of a hundred newtons and its angle it's east so it's on the x-axis so it has an angle of zero degrees the second force vector has a magnitude of 150 newtons and it has an angle of 30 degrees relative to the x-axis now what you want to do is you want to add these vectors using the component method so you want to break these forces into the components add their respective x and y components and then you could find the results in force vector so let's say if this is a force vector this is the x component and this is the y component and here is the angle f of x is equal to f cosine theta and f of y is f sine theta f you can find it by using pythagorean theorem it's f of x squared plus f of y squared and the angle theta is the arc tangent of f of y divided by f of x so those are some formulas that you're going to find useful in this portion of the video so what we need to do is find f one x and f one y so f one x is gonna be a hundred cosine of zero cosine of zero is one so a hundred times one is simply a hundred f one y is going to be a hundred sine of zero sine of zero is zero a hundred times zero is going to be zero now let's do the same for f2x and f2y f2x is 150 times cosine of 30 degrees now if you type that in your calculator make sure it's in degree mode so this will give you as a decimal 129.9 newtons f2y that's going to be 150 times sine 30 which is just 75. now what we need to do is add up the x components so if we add those two values the sum of the forces in the x direction is going to be a hundred plus 129.9 so that's going to be 229.9 and then we need to find the sum of the forces in the y direction which is just 0 plus 75 so that's simply 75. so f of x is 229.9 units and f of y is simply 75 newtons now let's go ahead and draw these values on an xy plane so f of x is 229 and f of y is 75. the resultant force vector is the hypotenuse and to find it it's going to be the square root of f of x squared plus f of y squared so you should get 241.8 so that's the resulting force vector that's the magnitude of it now that we have the magnitude we need to find the angle so it's going to be arc tangent f of x i mean f of y which is 75 divided by f of x which is 229.9 so the angle which is already in quadrant one is 18.1 degrees so now we have the magnitude and the direction of the resultant force vector