Consider the line through the origin in three dimensional space parametrized by t*v, where v=i+2j+3k. What is the distance from the point (-3,2,5) to this line? By the distance between a given point P and a line l, we mean the minimal distance between P and a point of l.

I was thinking that to get the angle between two vector (one is to get from the origin to the given point), and get the projection of vector a on vector b, so we could calculate the length of two side, and use the relationship to get the distance?

The line equation can be written as:



The distance from the point (-3,2,5) and the line:

Since (0,0,0) is on the line, we have





Cross product of u and v:


The distance is found by using the 'norms' of uxv and v:

If you don't remember the formula, you can still do it using the basics.

Make a quick sketch.

The line v has equation:



http://p1cture.me/images/42210695227422283694.png

The black line is the line v.

The length of the blue line is what you are looking for. Let the red line be l and the blue line be x.

Use the dot product of the line v and the red line, with vector <-3, 2, 5>:





Find theta.

You get the length of the red line from this formula itself, as you have to calculate the length of l.

Then use a simple trigonometric ratio:



And you get 4.44 units