Math Is Fun Forum

Reuel

Member

Registered: 2010-11-28

Posts: 178

Parametric value for the shortest distance from a point to a vector

Hello.

I need to find the value for t for the point on a line in space that is closest to another point in space. How is this done?

The specific problem: At what point in space does the vector-valued function

(being a line in space) pass closest to the point

?

My work thus far: I have used the formula

where P is the point on the line

to calculate that the shortest distance from

to the line

is a distance of about

.

The question I am having trouble answering is, what value for

on

IS this shortest distance?

I am stuck and appreciate your help.

Thanks.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,158

Re: Parametric value for the shortest distance from a point to a vector

hi Reuel

I'm not sure about your method.  I'd do it like this:

Let P be the closest point on the line.  Then PQ is perpendicular to the line. 

So if you form the dot product (or scalar product) between PQ and the direction of the line, this will be zero.

Form an equation from this and find 't'.

and

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Parametric value for the shortest distance from a point to a vector

Bob,

Thank you for your quick many-step response! I am working out what you have said and will get back to you.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,158

Re: Parametric value for the shortest distance from a point to a vector

hi Reuel

The numbers were horrible so I'm still looking to find the mistake.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Parametric value for the shortest distance from a point to a vector

Your answer is correct. If you plug that value in for t on the line and then calculate the length of the line segment from the point in space to the point on the line then you get my original answer for the shortest distance to the line, being 5.3753. What I didn't know how to do was to find the point on the line that this shortest distance from the point in space was.

You answered my question and now I see how to do it. Thank you. And if I have any further questions about it, I will return!

Thanks again.