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.
Let’s do another example. I want to plot the vector RP that starts at point R,(-2, 6) and ends at point P,(4, 9). I’ve got the points plotted here. Here’s R, here’s P. Let me draw a vector connecting them, starting at R ending at P.
I also want to find its length. Remember we do that by using the Pythagorean Theorem. So I have to draw a right triangle with a horizontal leg and a vertical leg. Here is the horizontal leg and I stop right beneath point P. I want to draw a vertical leg, and all I have to do is find the lengths of these two legs.
This one is 1, 2, 3, 4, 5, 6 units long. And this one’s 1, 2, 3, units high. The length of vector RP, written this way is the square root of 6² plus 3². That’s root 36 plus 9 which is root 45. Root 45 simplifies to 3 root 5.
What about this example; QS? A vector that starts at (-2, 1) and terminates at (4, 4). Let me draw that. Here are points Q and S. I’ll draw my vector, starting at Q and terminating at S. looks very similar to vector RP. Let me find this length.
Same way as before, I draw a horizontal leg and a vertical leg to make a right triangle. As I’m doing this, I notice that this is looking more and more like vector RP. Observe that this horizontal leg is exactly the same length as this one, so it’s got to be 6. The vertical leg is 1, 2, 3, same horizontal leg, same vertical leg. This means that the vector not only has the same length, but has the same slope. Since it has the same slope and it’s pointing in the same direction, these vectors have both the same direction, and the same length.
So vector QS has length 3 root 5. Remember when two vectors have the same length and same direction, they are equal. That’s really the only identifying characteristics of vectors; length and direction. These two vectors are actually equal. RP equals QS. We’ll talk more about vectors being equal later on when we do operations on vectors.