**The distance of a point from a line**

Vectors give us the tools to determine the distance of a point in space from a line in space.

A line in space passes through points *A (2,1,3)* and point *B (0,2,1)*. Determine the distance of point *C (1,1,1)* from the line.

The shortest distance is always the perpendicular distance from the line. Let *D *be the point on the line closest to point *C*.

We set the vectors

The vector *CD* can be described by the vector C*A* and the direction vector *a* of the line.

In addition, the vector *CD* is perpendicular to the direction of the line of vector *a*. So the dot product of the vectors is 0

Vectors *CA* and line direction vector *a* between points *A* and *B*.

Now we can describe the vector *CD* using these vectors

The dot product of the vector *CD* and the vector *a*

The dot product has to be* 0*

So vector *CD *is

The distance of point *C* from the line is the length of the vector *CD*

The distance of point* C* from the line defined by points *A *and *B* is *1*.

**Turn on the subtitles if needed**