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.
So here is a problem. It says point G has coordinates 5, 6 and 4. On this coordinate system, I've drawn rectangular box. The rectangular box's bottom sits right on the xy plane, flat on it. It's edge is backed up against the z axis.
Now, it says find the coordinates of A, B, C, D, E, and F. Those points are all labelled vertices on this box. Let's take a look at first of all how we get the coordinates of point G. Remember that 5 is its x coordinate, so that tells you how far out on the x axis it is. That means that this number here is 5. 6 is the y coordinate, that tells you how far along the y axis it is. So this point is 6. 4 is the z coordinate. That's how far up it is, 4. That will help us out.
Let's start with point A. Now point A is right on this point where x equals 5. Every plane on the x axis has a y coordinate and a z coordinate of 0. So this is going to be 5, 0, 0. Likewise, b has y coordinate 6, but its x and z coordinates are 0. So this is going to be 0, 6, 0. C is on the z axis. It will have coordinates 0, 0, 4.
Now let's get the in between points, starting with D. D is in the xy plane, and it's directly below point G. Because it's below, it's going to have a different z coordinate, but it will have the same x and y coordinates. So it's going to be 5, 6, and every point in the xy plane has z equals 0. What about E?
Well E is directly above point A, and it 's got to be as far above as point C is , or point G is, 4 units up. So it's going to be 5, 0 and 4. Similarly, F is going to be right above B. So it will be 0, 6, 4. That takes care of all six points.
Now let's do part B. Find the magnitude of vector OG. Vector OG is a position vector, starts at O, the origin, and ends at our point G. Now the components of OG are going to be exactly the coordinates of the terminal point G, so 5, 6 and 4. And so the length of OG is the square root of the sum of the squares of all three coordinates. So 5², 25 plus 6² is 36, plus 4², 16.
I'm going to get the square root of 36 plus 16 is 52, plus 5 is 57, plus 20 is 77. So square root of 77. That's it, that's the length of vector OG, root 77. So not only is that the length of vector OG, it also tells you how far point G is from the origin. The square root of 77 which is approximately 8.77.