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.
We're talking about the distance in Midpoint formulas in three dimensions. Let's do a problem. Find the length, and Midpoint M of segment AB. Now the length of segment AB is the same as the length of vector AB. So that's what I'm going to compute.
Now the formula tells me the square root of the difference in components squared. So I have 9 minus -3 or 12², plus 5 minus 1, or 4², plus 2 minus -4, 6². And that equals the square root of 144 plus 16 plus 36. Now 144 plus 16 is 160, plus 36 is 196. The square root of 196 is 14. So the distance between these points is 14 units.
Now what's the mid-point M? For the midpoint, you just average the coordinates. So -3 plus 9 over 2. 1 plus 5 over 2, and -4 plus 2 over 2. That gives me -3 plus 9, 6 over 2 is 3, 1 plus 5 is 6 over 2 is also 3, and -4 plus 2, -2 over 2 is -1. So the midpoint between these two is 3, 3, -1.
Let's try another one. I have points 3, 0, 4, and -1, 3, 4. Again I'll find the distance between them by finding the length of vector AB. That's the square root of the difference between these components squared. So -1 minus 3 is -4², plus 3 minus 0 is 3², plus 4 minus 4 is 0². So I have -4², that's 16 plus 9, that's 25. Root 25 is 5. So these two points are five units apart.
Now let's find the midpoint. Remember the midpoint is the average of coordinates. So 3 plus -1 over 2, 0 plus 3 over 2 and 4 plus 4 over 2. That gives me 3 plus -2, 2 over 2 which is 1, 0 plus 3 over 2, 3/2, and 4 plus 4 over 2 is 8 over 2 which is 4. So the midpoint between these two guys is 1, 3/2, and 4.
Let's take a look at a graph of the two points. So this is what we did in part B. A was 3, 0, 4, and B was -1, 3, 4. What's interesting about these points is that they both have a Z coordinate of 4. That means both of them are 4 units above the xy plane. So in a sense these two points lie on a plane parallel to the xy plane. The distance between them is 5, and the midpoint 1, 3/2 and 4. It makes sense that the midpoint has the same z coordinate.