The whole reason we use vectors in physics rather than complex numbers is that we need a third spatial dimension to describe directions in 3-dimensional space. It's also awkward to have one real axis and one imaginary axis when we're discussing topics like the position of an object in space. There is nothing more real or imaginary about any direction versus another. In this sense, even for 2D cases, vectors which have all real components make more sense than complex numbers.[br][br]Let's look at a situation requiring a 3D vector. In line with our original example of describing where you are relative to the college campus, consider the same situation except that you are flying nearby in an airplane. Now we'd need a third dimension to describe your location since altitude presents a third dimension. GeoGebra is able to plot in 3D. If you happen to have red/blue 3D glasses you can actually see the vectors in 3D otherwise you can just left click and rotate a vector as in the graphic below. Just to be clear, the [color=#ff0000]red is x[/color],[color=#6aa84f] green is y [/color]and [color=#0000ff]blue is the z [/color]axis. To see the 3D glasses view or to play around with an automatically rotating coordinate system, click the icon on the top right of the plot pane. There you can play around with options. Under the icon that looks like a cube there is a picture of glasses. That makes the scene 3D viewable with appropriate glasses.[br][br][color=#b45f06]AN ASIDE: [br]If you've never considered how graphics can be made to appear 3D, the whole trick is to send different images to each eye. Your eyeballs have a slightly different perspective of the real 3D world we live in. Your left eye sees a slightly rotated version of the world that your right eye sees, and vice versa. So we can make 2D screens look 3D by sending each eye a carefully rotated version of the same scene. Without special glasses this would mean seeing both images with both eyes. That would just look like double vision. With glasses, however, we can filter the image so only one of the images makes it to your left eye while the other makes it to your right eye. This is the same way 3D movies are made, except that they use polarized light and correspondingly circularly polarized lenses in the "3D glasses" you wear. The advantage polarization has over using red/blue lenses, is that both eyes can see a properly colored movie. The disadvantage is that it requires specialized projectors to send out polarized light, so this can't be done with an ordinary computer monitor for the sake of 3D graphics as in GeoGebra.[/color][br]

When we add a third dimension or component to the vector we can now describe locations or directions in 3D space. In our college example, our 3-component vector will now have an x-component describing east/west, a y-component describing north/south and a z-component describing up/down. By up/down I mean up toward the sky or down into the ground. [br][br]For instance, if you are 3km east of campus, 4km south (or -4km north) of campus and at an altitude of 1km above the campus, your position vector would be [math]\vec{r}=3km\hat{x}-4km\hat{y}+1km\hat{z}.[/math] This vector is shown in the graphic above. You can rotate the vector to see it from different angles. Just left-click and drag it. As mentioned, you can also make it auto-rotate and do other things as well. To change the values of the components, just change the values on the left where the vector is defined.[br][br]The total distance from campus to your position is given by the magnitude (or norm) of this vector, which is just the square root of the sum of the squared components, or a 3D Pythagorean theorem:[br][br][center][math]|\vec{r}|\equiv r=\sqrt{r_x^2+r_y^2+r_z^2}.[/math][/center][br]In the case of our present example, we get [math]r=\sqrt{(3km)^2+(4km)^2+(1km)^2}=\sqrt{26}km\approx 5.1km.[/math] Regarding the angle, we have a more complicated situation now. The reality is that there is no simple way to give a single angle to find you. We know that you are somewhere on the surface of a 5.1km radius sphere with the campus at its center, but where on it?[br][br]It would take at least two angles to specify your location on the sphere. Those two angles would need to be specified with respect to two different and known directions. These could, for instance, be with respect to the x and z axes as is commonly done in spherical coordinate systems. [br][color=#b45f06][br]AN ASIDE: For one value of angle, it would in fact be possible to use a single angle to specify your location with only one angle measured, for instance, with respect to the +x axis. Can you think what that value would be? While true for one angle, it is not generally true for any other angle.[/color][br][br]It is worth stopping and asking why both angles would be required. After all, in a 2D plane, if you know the direction with respect to the x-axis, then you also know it with respect to the y-axis. They are complimentary angles. The problem is that the same is not true for 3D space. [br][br]A simple way to illustrate this is to ask about the angle between some other direction and the x and y axes. One choice would be the z-axis itself. Obviously it is 90 degrees between the x-axis and the z-axis, but it is also 90 degrees between the y-axis and the z-axis. These are clearly not complimentary angles. It turns out that the sum of the angles between some random direction and the coordinate axes is not a fixed value.[br][br]The most common thing to do then, is to find the angles between a vector and the coordinate directions using something called direction cosines which we'll discuss in the next section.[br]