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Three-Dimensional Vectors

I. The Three-Dimensional Coordinate System

We will look at 3-d vectors mainly in component form, and so we first need to define the 3-d coordinate system. We still have the x- and y-axes, of course, and to create our third dimension we add a z-axis. It is customary, however, to shift our view of the x- and y-axes, by letting the x-axis come "straight off the page" and defining y as the horizontal axis. This leaves the z-axis as the vertical scale, as shown on the left.

Because of the physical difficulties of plotting three-dimensional objects on a two-dimensional surface (your screen), the actual coordinates of points (and thus the components of vectors) can be rather vague if we don't use labels. Fortunately, using Maple we can always adjust our view to get a better sense of what's going on, as in the rotating graph below where we see vectors

and

from several angles.

II. Operations on Three-Dimensional Vectors

All operations defined for two-dimensional vectors are also defined for three-dimensional ones: we can add, subtract, scalar-multiply and dot-multiply vectors. We can still find a vector's magnitude, only now we must take three coordinates into account; in the same way, the dot product will use two trios of coordinates instead of two pairs.

For our vectors on the right, some simple examples would be

And we can still find the angle between the vectors using our dot product formula:

Click here for animation code

As with our two-dimensional vectors, Maple is well-equipped to perform all these functions:

III. The Cross Product

With our extension into three dimensions, we now define a third type of multiplication: the cross product (sometimes called the "vector product"). The notation is u x v, which is read "u cross v," and the result is a vector that is perpendicular to both u and v (see the diagram). The general formula for calculating the cross product of

and

makes use of a 3 x 3 determinant:

where vector i (1,0,0) is the unit vector along the x-axis; vector j (0,1,0) is the unit vector along the y-axis; and vector k (0,0,1) is the unit vector along the z-axis. It is straightforward to show (using the dot product) that u x v must be perpendicular to both u and v.

For our example vectors, the cross product would then be

which is plotted in blue above. Again, Maple has a function to avoid the tedious calculations:

IV. The Area of a Parallelogram

One of the properties of u x v is that its magnitude is equal to the area of the parallelogram defined by u and v; for our example vectors we would then have

And this is easily confirmed as the area of the parallelogram:

Problem Set

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Three-dimensional Vectors