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.
All we need to find a vector is to get its initial point and its terminal point. Here I have vector QP, and I’m asked to draw the vector and compute its magnitude. Point Q is (-2, 1) and point P is (4, 9). I plot this vector, starting at point Q and terminating at point P. All I have to do is draw an arrow connecting the two, so staring at point Q and ending at point P. Very easy to draw a vector when you’re given the initial point and the terminal point.
How do you find its length? It’s easy, just use the Pythagorean Theorem. I’m going to draw a right triangle where the hypotenuse is QP. I draw a horizontal leg and I got to stop right beneath the point P, and a vertical leg. To find the length of the vector, I have to find the lengths of the horizontal and vertical legs.
The horizontal leg is 1, 2, 3, 4, 5, 6 units and the vertical leg is 1, 2,3 4, 5, 6, 7, 8, units. The length, and remember that the length is written like absolute value. Absolute value of QP is the square root of 6² plus 8². This is the length of the horizontal leg. This is the length of the vertical leg. That’s the square root of 36 plus 64. That’s the square root of 100 which is 10. The length of vector QP is 10.
Let’s do another example. Let me plot vector SR, which starts at point S(4, 4) and terminates at point R(-2, 6). Here is point S, here is point R. Remember the vector starts at S and ends at R. It’s a different vector that starts at R and ends at S. I put a little arrow head at R, that’s the terminal point.
To find its length, I’m going to use the Pythagorean Theorem again. I need to draw a right triangle with a horizontal leg, like so, and a vertical leg. This horizontal leg is exactly as long as the horizontal leg I used for the vector QP, it's 6. The vertical leg is 1, 2 units high. To find the length again I use the Pythagorean Theorem.
The length of vector SR is the square root of 6² plus 2². That’s 36 plus 4, the square root of 40. Now to simplify this, what’s the biggest perfect square or factor of 40? I think it’s 4. So this is going to be 2 root 10 and that’s my answer.
Remember when you’re graphing vectors, start with the initial point, end with the terminal point. The initial point is the first letter in the vector notation; the terminal point is the second letter. And when you’re finding the length use the Pythagorean Theorem.