**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

(1)

We want to find a line passing through these points. What to do?

First we form the vector

(2)

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

(3)

and this is equal to some scalar multiple of (i.e., it’s a dilation of the vector). We write

(4)

which lets us write

(5)

This is the parametric equations of . So returning to our example, we have

(6)

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

(7)

This is all abstract, lets consider an example.

**Example 1.** * Find the distance from to the line given by *

(8)

First we pick the point when , we call it

(9)

Observe

(10)

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

(11)

Its unit vector is

(12)

We have

(13)

This has its magnitude be

(14)

which describes the distance between our point and the given line.

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## About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.