Vectors and Parametric Equations
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We’re talking about vector addition, and I want to do an example that will show you that vector addition is commutative, which means that the order that you add 2 vectors doesn’t matter. RS plus QP will turn out to equal QP plus RS. But let’s see that and see why that’s true.
First of all R, S, Q and P are all defined by these coordinates. I’ve plotted them over here. So you can actually see vector RS and vector QP. Now I want to add RS and QP first. So I’m going to start by leaving RS in place, and move vector QP so that the two vectors are head to tail. The way I do that is I make note that vector QP is 6 units to the right, 8 units up. That’s basically the definition of vector QP; go 6 to the right, 8 units up.
So when I draw QP, when I draw a copy of it, starting at point S I got to go 6 across and 8 up, like this. And that will be an exact copy of vector QP. Now the sum of the two vectors, is going to start at point R and it will end at the terminal point of QP. So it looks like this. So that’s RS plus QP.
Now let's find its length. We can find its length by drawing a right triangle, and since I’ve got a lot of vectors here I’m going to draw the right triangle above. So I’ll draw the vertical leg going up this way, the horizontal leg going across the top. What’s the length of the vertical leg? We have 1, 2, 3, 4, 5, 6, and the horizontal leg is 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. The length of RS plus QP is the square root of 6² plus 12². That’s 36 plus 144. Let’s make the observation that 44 is 12 times 12, this is 3 times 12, so this is 15 times 12. Do we have any perfect square factors here? We’ve got a pair of 3's, we’ve got a 4, so I’ll move one of the 3's over and I get 5 times 36. And you can see that this is going to end up being 6 root 5. That’s the length of RS plus QP.
Now let’s look at QP plus RS. When I draw this sum, I’m going to leave vector QP alone and I’m going to move RS so that it is head to tail with QP. This guy is going to have to move up and over. Now vector RS goes down 2 and to the right 6. When I duplicate it, my new vector has to go down 2 and 6 to the right. This will be my duplicate.
The sum of those two vectors, I draw by starting at point Q and ending at this point. This is my sum. The two vectors, the two sums, this is RS plus QP, this is QP plus RS, they look exactly the same. But the real test is do they have the same length and do they have the same direction? Actually when I draw my right triangle, I’ll have an idea of what slope this vector is, and that will give me an idea of the direction. Let me draw that right triangle.
Vertical leg and let’s find the lengths. This is 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Just like before. And vertical, 1, 2, 3, 4, 5, 6. 6 like before. Exactly the same triangle. This is going to have to be the same length and the same direction. These two are truly the same vector.
Vector addition, the order doesn’t matter. Vector addition’s commutative. Now since I have the same exact triangle I can conclude that the length is also 6 root 5.
The property in mathematics when the order of an operation is reversible is called commutativity. Vector addition like real number addition, is commutative.