1. We will use our knowledge of vectors to construct lines in 3-space, i.e., three-dimensional Euclidean space.
2. Constructing Lines. Suppose we have two points
We want to find a line passing through these points. What to do?
First we form the vector
This vector is parallel to ; the numbers given by this vector’s components (i.e., -1, 3, 8) are called the “Direction Numbers” of .
In general, we have two distinct points and on the line , then we construct the vector
and this is equal to some scalar multiple of (i.e., it’s a dilation of the vector). We write
which lets us write
This is the parametric equations of . So returning to our example, we have
where the constant terms are precisely the values of the components of , and the coefficients of are the components of the vector .
3. Distance From a Point to a Line What’s the distance from any point in 3-space to a given line ?
We pick a point on and form a vector . The distance from to can be given as , as in the following diagram:
We see . If is a vector parallel to , then we have
This is all abstract, lets consider an example.
Example 1. Find the distance from to the line given by
First we pick the point when , we call it
Now we need to find a vector parallel to the line. What to do? Construct a vector by considering the point when , which would be . Thus
Its unit vector is
This has its magnitude be
which describes the distance between our point and the given line.