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Vector Addition Video #1

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This video introduces vectors and explains how to add vectors that are parallel or perpendicular to each other.

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hello AP Physics today we're going to talk about vectors and beginner discussion on vector addition all quantities in physics be divided into two categories a vector or scalar quantity scalar quantities are quantities Express completely by using numbers and units for example 5 seconds while Vector quantities require a number and unit as well as a Direction to complete their description please note the word magnitude on the screen a number and a unit put together can also be called the magnitude of the vector  that when we're using Vector quantities throughout physics you're going to see them represented in formulas you can identify a quantity pretty easily by noting that the vectors either have an arrow over the variable or it's in bold Sprint for example the formula  force is equal to mass times acceleration has to Vector quantities force and acceleration so to correctly represent this we would put arrows  over the force and acceleration or it can be written if your horse is in bold acceleration is in boldface text two ways to represent Vector quantities when writing formulas also whenever we vectors in our answers we always want to make sure we put the direction in Brackets and finally doing Vector diagrams use an arrow to represent the direction The Links of the arrow will represent the magnitude of the vector in the direction that the arrow points will indicate the direction of the vector  some more examples of vector and scalar quantities scalar quantities would be adjust Time 5 Seconds distance 3 metres speed 15 miles per hour in Mass 3 kg well all Vector quantities also include a direction such as displacement 3 meters East velocity 50 what miles per hour north  Force 10 Newtons up in momentum 25 kille grams meters per second down please take a look at your vector Packard packet that you've been given or you can get how's the web page and noticed that the very first question is asking you to draw and write vectors the first one we have here is to draw a vector that represents 15 meter South when drawing vectors we want to start with a Cartesian plane notice the Cartesian plane can so represent an aerial view of a map north south east and west if we want to draw a vector that's 15 M South you're going to start at the origin and draw an arrow pointing South however long you want it to be can represent 15 meters cuz it's going to be a rough drawing so put the quantity and the unit and that's how you represent this meter self question on your packet ask you to write the vector that's represented by the drawing so if I had a vector at this point you can imagine the Cartesian plane  that kind of goes over this and you can see that the vectors pointing to the so you represented as 12 meters that's the magnitude bracket to the east and that represents the Quan shown  when we have two or more vectors we can add them together According to some rules called vector addition but there's new methods that we could use to add vectors up the first is called the graphic method of the graphical method and that is when you have to draw your vectors scale using a ruler and a protractor along with graph paper you don't need any calculators or really need any knowledge of your trig to get this answer now we're not going to use this method in class however I will expect you to know what the method is and how to apply it if you ever need it to the second method is the trigonomic or the mathematical method we still make a drawing here a rough diagram but we're going to use are pre-cal trig functions sine cosine tangent as well as of that green theorem to help us solve some answers well I'm going to lawn an animation and it's going to show two vectors being added together you can see that there's going to be a blue vector and a green vector and they're going to add to weather in are going to produce the resultant Vector which is The Black Arrow please notice or note the orientation of the blue and directors as they're adding together and what the result is with the black Vector that's shown okay take a minute and make some observations notice but when the blue Vector in the green Vector are both pointing in the same direction that the resultant vector or the black vector at its maximum length  when you have two vectors that are in the same direction you simply add the vectors magnitudes and you keep the of the vectors for your answer another thing I want you to take a look at our the angles there any time you have two vectors that are pointing in the same direction the angle between those vectors is either zero degrees or 360 degrees anytime two vectors are have an angle 0 degrees between them it'll produce the largest magnitude Vector possible in that situation  no no but winda blue Vector in the green Vector are pointing in opposite directions your resultant or black Vector as is at the shortest or smallest value this is when the two vectors are 180 degrees apart  and to do vector addition in that scenario you simply subtract what is actors and keep the direction of the largest vector that we know how to produce the largest possible deck in the smallest possible doctor and we also should realize now that any other Vector can be produced with angles between 0 in 180 it will learn how to do that  throughout the next week this next animation is going to show you a property a vector addition that relates to some old math properties that you probably realize we added five vectors together Becker ABCD and e but we added weather in a different order note to the resultant Vector which is the green Vector that's produced it all three cases there Vector has the same magnitude and is pointing in the in the same direction so the order that you add the vectors will not matter and doing vector when drawing Vector diagrams to do tradition you're always going to start with the diagram we're not going to do it to scale so you can make a rough diagram if the vectors are in the same direction add the numbers and keep the direction you're the vectors are in opposite directions subtract the numbers and take the largest Direction all other cases can I use a rough diagram and use a parallelogram method along with your trigonometry to solve for your answer now this is a pretty simple flowchart that were going to use throughout the next week we're going to take it piece-by-piece here real you're going to start with the vectors that you're given and you're going to ask yourself the question are the vectors given parallel that means are pointing in the same direction where are they opposite if you say yes to that then you're going to Simply add or subtract the vectors based on if they're going in the same direction opposite get your direction and that's your answer  the first page of your vector addition worksheet is to deal with vectors that are parallel let's take a look at example of that  dr. a is 7 km per hour north Vector B is 5 km per hour in North Vector see when Vector C is equal to Vector a plus deck Derby you can find this example on your packet if you want to follow along since both of my vectors are in the same direction we're simply going to add our numbers so 7 + 5 is 12 we're going to keep our units and a direction Remains the Same so our final answer would be 12 kilometers per hour north  the second example that's provided on your in your packet since Vector a is 120 M West Vector B is 12 m east what's decter see when jector C is a + b noticed that your directions are opposite so you're going to subtract your magnitudes and you'll get a hundred and eight  meters and since 120 is larger than 12 we're going to keep the direction  West you're now signed before your turn problems in your packet  now let's go back and look at the flow chart and see what happens if we have to say no are vectors that were starting with are not parallel then the next thing I want you to check is to see the vectors provided are perpendicular to each other which means there has to be a 90° angle between those if you answer yes to this you're going to make your diagram and then you're going to use your trig functions that you learned in in precal to help you solve this problem but before we get to the breakout I want you to know that in your textbook chapter one Turn 5 figure 1.23 has a really nice review of your trigonometry functions that you learned in pre cal notice when you have a triangle is shown here you have an angle represented the side opposite the angle is represented with an O fried adjacent to the angle represented is represent with an a in the hypotenuse is always opposite to the right angle and Euler 3 trig functions sine cosine and tangent you might have remembered sohcahtoa from an earlier math class the Sun of any angle which me the scientist angle would be equal to the side opposite divided by the hypotenuse cosine of any angle is equal the side adjacent divided by the hypotenuse while tangent of an angle is equal to the side opposite divided by  the adjacent well we will work extensively with degrees and will not be working with radians while dealing with vectors and vector addition  another notable formula mass situation that you learned was the Pythagorean theorem and it's going to come in handy here remember that a squared plus b squared is equal c squared you can always find the length of the hypotenuse by taking the square root of the other two sides squared and then add those sometimes you'll have to work backwards which is shown in this last call last two columns hear that if you know the angle or so if your team for the angle you might have to work backwards and use the inverse sine  the inverse cosine or the inverse tangent when you're solving for the let's go ahead take a look and an example problem and this is in your packet you can follow along on the back of page 1 Victory is 122 km per hour north inductor B is 52 km per hour West find the resultant vector you can kind of visualize this as if you had a plane traveling 122 km per hour north and then there was a crosswind of 2 kilometers per hour West you're playing would go off course and we can find the direction of the plane by doing vector addition the first thing we're going to make sure that we do is  to draw are vectors a rough diagram there's our Cartesian plane bad if we draw are vectors using the parallelogram method we start all vectors from the origin in a backup can identify this north south east and west so now the origin which is 100 22 km North you can make that Vector as long as you want  well you probably want to give it a little bit of space so that you can work and everything's not real cramped and tight now we need to drive Vector to be which is 52 km per hour West the parallelogram method requires us to draw all of our vectors from the origin so our second vectors going to be 52 meters or suming 52 km per hour in the west erection  about what we're going to do next is we're going to make a parallelogram out of the two actors that are drawn so I'm going to just with the dotted line show the parallelogram would be produced by those two perpendicular vectors  now in a different color I'm going to use green we're going to represent resultant Vector to do this you go back to the origin and you draw your resultant Vector from the origin and you draw a diagonally across your parallel the green Vector shown is going to be our vector and they were looking for and we're going to call it Vector see the first thing you want to do is find the magnitude of this vector and hopefully what you'll recognize now is that we can use the Pythagorean theorem to find the magnitude The Links of our green Vector looking at this triangle right here we would see that the bottom side is 52 the side has to be 120 since it is a parallelogram and we can find that Vector see  is equal to the square root of 52 squared + 122 squared if you plug your number into the calculator you should get the the magnitude will be 132.6 kilometers per hour using significant fig weekend that to 130 kilometers per hour now that only provides a the magnitude of our Vector we also need to give the direction and simply Northwest will not be descriptive enough so we're going to go back torque diagram and we have to represent which angle we want to measure we always measure angle close to the origin as well and as you can see we have to a lower angle angle and a top angle it doesn't matter which one you select but you have to mark it on your diagram in order for your. For your work to be correctly graded we're going to call that angle and I'm going to call an angle Theta  Define Sada we now have all three sides of our triangle notice we've got the side at the bottom 120 or 52 you've got the side  going flip 122 and we just found the hypotenuse which we would use as 1:30 to 6 because we wouldn't want to use a rounded value yet so we can use our sign tangent  trig functions to help us out here and although any of the three would work the trig function jet  sees me the trig function tangent Theta the side opposite over the side adjacent would work best here because we're dealing with information provided in the problem instead of the hypotenuse which we calculated no notice what we want is only Theta not tangent Theta so we need to take advantage of the information I talked about a few minutes ago that deals with this column when we're looking for the angle  so in order to correctly solve for angle Theta we would need to take the tangent to the other side which means it becomes Envy arctangent of opposite over  adjacent that we need substitute in  her values inverse tangent of the side opposite of our triangle would have the 122 decided Jason to the angle that we marked was 52  and note that on the calculator what are in my class you need to do what's in parentheses first and get 122 / 52 once you get that value you didn't take the inverse tangent of that and you'll end up with an angle of 66.9 which will always full angles which will round to 67 degrees so to put this whole answer together we would now say that are resultant vector or vector C is 130 kilometers per hour bracket 67 breweries and listen to how we have to represent it from here this angle that we drew was 67 degrees north of the West access so we would say north of the West  access and that's how we represented the answer or back to the airplane the airplane would be traveling with a with a velocity of 130 kilometers per hour can a directions 67 degrees north of West what happens if I had decided to find the other angle I told either one would work sufficient at so what happens if I wanted to Orange angle that I'm going to call  Alpha what would have happened is when we took our inverse tangent our side opposite now would have been 52 and our side adjacent then 122 and when we did this calculation we would have ended up with the angle being 23 degrees but a quicker way Define the other way to represent this answer would be to Simply take 90 degrees minus the 67 that we got earlier and that give us the complementary angle or 23 degrees so another way to write this answer which would have been correct could have been 100 30 kilometers per hour at 23 degrees but now you would have had to say the angle is 23 degrees west the north axis notice that your directions swapped places  so there's two correct answers for all vectors when you're doing vector addition but you need to make sure you know which one you're actually trying to solve for and be able to find the other one in case that's the answer on a multiple choice quiz or test don't want to let you know that your homework for this evening is going to be the front and back of page 1 your vector addition handout and you're going to have a quiz and then next class another note I like you to be aware of is that there is going to be Vector assignment in the mastering physics online that will be due in a week and you need to be start working on that next week when we do some more india-west situations  a scenario three this point you should know how to saw Vector problems for one and two after we take our quiz next class. We're going to look at what happens if the vectors are not tell a perpendicular and we'll deal with that next time hope this video has helped have a great night

Uploaded by Nicole Comstock.

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Vector Addition Video #1

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