Unit
Vectors and Parametric Equations
Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
The first operation we learn with vectors is addition. Adding vectors can be done two ways with the geometric representation. Both of these methods help us understand what it means to add two vectors. When the vectors present forces, adding vectors is called finding the resultant. The algebraic representation of vectors has a different method of addition.
I want to talk about vector addition and to do this I'm going to start with a story. Slappy the dog starts at his home a and runs to pick up his girlfriend at her house b 9 blocks east and 6 blocks north of his home. Draw a vector ab representing his displacement. Let's do that, now he's gone 9 blocks east and 6 blocks north so I'll make that 9 and that 6 and then his vector his displacement vector will be this and to complete the picture I should probably note that this is point a, his home, and this is point b his girlfriends house.
Alright moving on, then the two of them run 4 blocks south and 7 blocks east to Cartuni's point c where they share a plate of spagetti with meatballs. Draw vector bc representing their displacement. So they've gone 4 blocks south and 7 blocks east, let's say that, wrong pen, 4 blocks south is to here and 7 blocks east something like that, so the vector that I'd want to draw is this and this is point c. This is vector bc their diplacement from her house to Cartuni's.
Finally the resultant or sum of these two vectors ab+bc is the vector ac and this represents slappy's displacement from his house. He's 16 blocks east and 2 blocks north. Let's take a look, if you think about where his location is at point c with respect to his original location at point a he is 16 blocks [IB] east and 2 blocks north and that vector would be vector ac, so we define this vector to be the sum of the 2 vectors ab and bc so this is vector ac and ac is ab+bc. Now this suggests how we're going to think about vector addition. We're going to think about vector addition first as taking two vectors and placing them head to tail like this and drawing what we call the resultant from their the initial point of the first vector to the head of their second vector so let's see that in an example.
Here I have two vectors u and v, if I want to add them I have to put them head to tail so say I want u+v, I can take this vector and slide it over so that its head is right at the tail of vector v. You can always slide vectors around because two vectors are equal as long as they have the same length and direction so let me draw a vector with the same length and direction as u and it will be this right, as long as it has the same length and the same direction it's the same vector and so the sum of these two vectors could be the vector that you draw from the tail of the first to the head of the second that's vector u+v so one method for adding vectors is the head to tail method you put the vectors head to tail and you draw the resultant from the tail of the first to the head of the second.
The second method it's the parallelogram method and here is how that works. We start by taking the vectors and putting them tail to tail so let me duplicate this vector down here it will look like this, okay where here we go, so that's vector [IB] now imagine that these are two sides of a parallelogram and you want to complete the parallelogram. The way you do this is you duplicate this vector down here and duplicate this vector over here, so that looks like this and this vector goes down one over 1, 2, 3, 4 I'm counting each of,I'm counting two squares as one unit here so down one over 1, 2, 3, 4 and the other vector looks like this and so if you draw a vector starting at the tail of the of the two vectors and ending at this point you'll also get the sum. This is the parallelogram method and even though it's a little a little more time consuming, it's very useful in certain applications but this also is u+v. So we have two ways of coming up with the sum of two vectors; First the head to tail method, put the vectors head to tail and it turns out that vector addition is commutative so the order you do this does not matter and then once you get them head to tail, draw the resultant or sum as the vector that goes from the tail of the first to the head of the second or put them tail to tail complete a parallelogram and draw the vector that goes from the tails the common tail to the opposite point of the parallelogram.
These are the two methods for adding vectors Geometrically.