Thread: Point on Circle Closest to Another Point on Circle

Hi all ---



I understand the circle equations. And I understand the vector joining the two centers of the circles is just ---
so its norm is

But I don't understand the in the last line at all. Where does it come from? What does it mean?

Thanks a lot ---

distance between the two centers is 5

radius of the first circle is 2

point on the first circle closest to the second lies 2/5 of the total distance from center to center.

start position (center of the first circle) + 2/5 of the direction vector from center to center

(5,4) + (2/5)<-4,-3>

Originally Posted by skeeter

distance between the two centers is 5

radius of the first circle is 2

point on the first circle closest to the second lies 2/5 of the total distance from center to center.

Hi skeeter ---

Thanks for answering. The part above I get. But the rest of your post I don't, sadly.

I drew a sketch ---





Because = vector to the center of the large circle is NOT on the same line as = distance between the two centers ---

how does [ give us the point we want?

I get your post if the green and red lines are collinear, but that's not the case here?

Thanks a lot ---

start position (center of the first circle) + 2/5 of the direction vector from center to center

(5,4) + (2/5)<-4,-3>

think of it as adding two position vectors ...



the sum of the two vectors is the position on the large circle closest to the center of the small circle.

Hi skeeter ---

Thanks so much for answering.

I get how to do arithmetic with vectors, but I still somehow don't understand why works, probably because and are NOT collinear.

Here's what I'm thinking --- gives the radius of the big circle right on - the vector joining the two circles's centers.

But doing looks to me like we're getting the part/rest of the red vector that's outside the big circle. But the red vector isn't on - so it wouldn't give the point on the big circle closest to the small circle?

And I know that the part of the green and red vectors inside the big circle have the same length in the big circle, because they are both radii.

Originally Posted by mathminor827

I get how to do arithmetic with vectors, but I still somehow don't understand why works, probably because and are NOT collinear.

The set of points is the line segment between the centers.
When we get
When we get
When we get the point we want.

Originally Posted by Plato

The set of points is the line segment between the centers.
When we get
When we get
When we get the point we want.

Hi Plato ---

Thanks for answering. But aren't you working backwards from the answer? I mean - I understand what you're saying.

But I'm stuck on the idea or process to actually get the answer - I'm not asking about the actual values of the coordinate.

Thanks ---

Originally Posted by mathminor827

But aren't you working backwards from the answer? I mean - I understand what you're saying.

Absolutely not!
The length of that line segment between centers is 5.
The radius of the circle centered at (5,4) is 2.
From we want a point that is the way between the centers.
What is backwards about that?

centers of circles 5,4 and 1,1 give a slope diagram delta y =3 delta x=4. hypo =5. the point required is three units from 1,1 along the hypo. By similar triangles 3/5=h/4 h=12/5 x= 1+12/5 =17/5 =3.4
3/5=v/3 v=9/5 y= 9/5 +1 =14/5 = 2.8