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 have one important thing that we have to do. We have to define how to find the components of a vector that goes from one point Q, to another point R, and also find its length. Now, we've already done this for position vectors; vectors that start at the origin, and end at some point. So the way I want to go about dealing with the components of QR, and it's length, is I want to translate this vector down to the origin.
Now you remember that two vectors are equal as long as they have the same length, and same direction. So if I can translate this without changing it's orientation at all, I'll have exactly the same vector. I'll just have a position vector version of it. So here is the idea.
I want point Q, the initial point of the vector, to be translated down to the origin. So what I'm going to do is subtract x1 from the x coordinate. Subtract y1 from the y coordinate, and subtract z1 from the z coordinates. I'll get 0, 0, 0, so I'll get this point.
Now the way I get R down to here is I subtract exactly the same things. I subtract x1 from the first coordinate, y1 from the second, and z1 from the third, and I'll get this point. So the coordinates of P are going to be x2 minus x1, y2 minus y1, and z2 minus z1.
So these two vectors will be exactly the same. That means that QR is going to equal OP. The components of OP are exactly the coordinates of this point. So they're going to be x2 minus x1, y2 minus y1, and z2 minus z1. Now how I find the length, well I know how to find the length of a position vector. So I'll just find the length of the position vector OP, and that will the length of QR.
So the length of QR is the length of OP. That's the square root of the sum of the squares of the components. So I have x2 minus x1 squared, plus y2 minus y1 squared, plus z2 minus z1 squared. So these are two very important results. They're basically about one thing; about vectors going from one point to another. Any one point is space to any other point in space.
So QR in component form is x2 minus x1, the difference in the x coordinates. Y2 minus y1, the difference in y coordinates, and z2 minus z1, the difference in z coordinates.