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.
With vectors, we begin to work more with the 3D coordinate system. In the 3D coordinate system there is a third axis, and in equations there is a third variable. We will work with vectors in the 3D coordinate system and learn how to interpret the coordinates an of a vector in the 3D coordinate system. With the introduction to the 3D coordinate system, we also encounter other vector operations, lines and planes.
I want to talk about the 3 dimensional coordinate system. Here I have a picture of a 3 dimension coordinate system it's got 3 axes, the x axis, the y axis and the z axis. And this is the typical orientation with the x axis sort of coming out at you, the y axis going to the right and the z axis going up. So the x, y plane that you're used to is lying flat, now how do you plot points in a 3 d coordinate system? Let's start by plotting point a which has coordinates 1, 2 and 5 since there are 3 axes we need 3 coordinates and it's going to be x, y, z, 1, 2 and 5. So first I look for x=1 and that's here and then y=2 and that's here and I find the point in the x, y plane that they correspond to and that's right here. And then for the third coordinate 5 I just go up 5 units, so this is unit I go 1, 2, 3, 4, 5 and this is going to be my point a.
Let's try another 1, b which is 4, 6, 2 first thing I do is I locate 4 on the x axis that's 1, 2, 3 4 I locate 6 on the y axis 2, 3, 4, 5, 6 is right here so four 6 is down here right on the x, y plane. Now from here I need to go up 2 units this is how long a unit is so I go vertically up 1 and then 2 and that's going to be my point b which is 4, 6, 2. So that's basically how points are plotted in a 3 d coordinate system. Now let's talk about vectors in the coordinate system, specifically positioned vectors now remember in 2 dimensions a position vector is a vector that goes from the origin to a point p and it's exactly the same thing in 3 dimensions. So let's draw the position vector o, p goes from the origin to point p which has coordinates x, y and z.
Now I want to find, I want to be able to write the components of a vector like this and the components of a position vector will always be just x, y and z the coordinates of the point where vector terminates. Now I want to find the length of this vector op and I can do it using what I know about 2 dimensional vectors. Let's talk about the point that is right below it in the x, y plane point a. Point a is going to have coordinates x, y and 0 right all the points in the x, y plane have a z coordinate of 0. So this will be x, this will be y and this tells me that, this distance here can be found using the Pythagorean Theorem what that distance is, is the length of vector oa. So oa its length squared is going to be x squared plus y squared, x is this length and y is this length so x squared plus y squared.
Now secondly let me take a look at vector ap, ap goes straight up from point a in the xy plane to point p and its length is z. So the length of op squared, the length of this length squared is going to be this length squared plus this length squared because this is a right triangle and that's hard to see when you have it drawn in a perspective like this. But this is a right angle, there's any vector pointing straight up is going to be perpendicular to any vector lying flat in the xy plane. Alright so is going to be equal to oa squared plus ap squared. Now we already found oa squared it's x squared plus y squared and ap squared is just z squared. And here we have it, the magnitude of op squared is x squared, plus y squared, plus z squared. So the magnitude of op is the square root of x squared plus y squared, plus z squared.
This magnitude not only gives you the length of the position vector op it also gives you the distance from this point to the origin. So the distance from point p to the origin is x squared, the square root of x squared plus y squared, plus z squared.