# Use vectors to form a right triangle on a circle

ibaforsale

## Homework Statement

Use vectors to demonstrate that on a circle any two diametrically opposed points along with an arbitrary third point(on the circle) form a right triangle

## Homework Equations

Hint: assume without a loss of generality that the circle is centered at the origin and let v, -v, and w denote the three points in question. show that the vector connecting w to -v is orthogonal to the vector connecting w to v

## The Attempt at a Solution

i think i have a grasp on how to achieve this. to show that they are orthogonal the dot product of the two vectors must be 0.

I am confused with the v and -v. In my mind that makes a straight line for example say that v is (1,0) then -v would be (-1,0).

I dont see how those two points along with a third make a right triangle.

Mentor

## Homework Statement

Use vectors to demonstrate that on a circle any two diametrically opposed points along with an arbitrary third point(on the circle) form a right triangle

## Homework Equations

Hint: assume without a loss of generality that the circle is centered at the origin and let v, -v, and w denote the three points in question. show that the vector connecting w to -v is orthogonal to the vector connecting w to v

## The Attempt at a Solution

i think i have a grasp on how to achieve this. to show that they are orthogonal the dot product of the two vectors must be 0.

I am confused with the v and -v. In my mind that makes a straight line for example say that v is (1,0) then -v would be (-1,0).

I dont see how those two points along with a third make a right triangle.
That's the object of this exercise. Take any arbitrary point w on the circle, and form displacement vectors between it and v and -v. What are the coordinates of w?

ibaforsale
can w be (1,1)?

Mentor
No. That point is not on the unit circle, which I assume is the one you're working with since you are using the points (1, 0) and (-1, 0). You need the equation of whatever circle you're using so that you can write the coordinates of your third point on the circle.

ibaforsale
my apologies, lets make w (0,1). so then I see that if you connect the lines you get a right triangle. so then to show that I must show that wv is orthogonal to w-v.

so for wv it would be v - w = <1,-1>
and w-v would be -v - w = <-1,-1>

then i take the dot product and if its 0 theyre orthogonal meaning they do form a right triangle, is that right?

Homework Helper
Gold Member
But the problem says that for any point v and its opposite on the circle, and any third point w that the vectors are perpendicular. You can't just do it for (1,0),(-1,0), and (0,1).

ibaforsale
could it just be with general terms?
v = (x1,y1)
-v = (-x1,-y1)
w = (x2,y2)

Homework Helper
Gold Member
could it just be with general terms?
v = (x1,y1)
-v = (-x1,-y1)
w = (x2,y2)

Yes. What do you get when you use those. Remember they are on the circle.

ibaforsale
So for wv = v-w = (x1-x2, y1-y2)
and for w*-v = -v - w = (-x1-x1, -y1-y2)

after doing the dot product i end up with

-x12+ x22 - y12 + y22

but then thats not 0

Mentor
So for wv = v-w = (x1-x2, y1-y2)
and for w*-v = -v - w = (-x1-x1, -y1-y2)
It should be (-x1-x2, -y1-y2)
after doing the dot product i end up with

-x12+ x22 - y12 + y22

but then thats not 0
You're not using the fact that all three points are on a circle. What's the equation of your circle?

ibaforsale
sorry that was a typo, i copy and paste to make the sub and sup easier

the equation of the circle is x2 + y2 = r2 i believe

Mentor
OK, so what can you say about the expression you ended up with for the dot product?

ibaforsale
well if x1 = 1 and x2 = 1 then that would cancel out and same for y but im not sure if that would work in the general sense.

Mentor
No, that won't work. Your point w (x2, y2) is a point on the circle whose equation is x2 + y2 = r2, right? So are v and what you're calling -v.

What characteristic do points on this circle have that all other points don't have?

ibaforsale
all the points on the circle will have the same radius

Mentor
That's true, but irrelevant. The radius of a point is 0.

Let me see if I can make this simpler. Consider the line whose equation is 3x + 2y = 6. Is the point (2, 1) on this line? Why or why not? Is the point (0, 3) on this line? Why or why not?

ibaforsale
(2,1) isnt because when you plug it into the equation it doesnt equal 6, (0,3) is because it does equal 6. so then if they are all on the same circle they would be the same when plugged into the equation, is that what youre hinting at?

Mentor
Yes. Any point on the line has to satisfy the equation, and any solution (a pair of numbers) to the equation represents a point on the line. Same for points on the circle.

With that in mind, what does that say about the value you got for your dot product back in post #9?

ibaforsale
im not entirely sure, im a little thrown off because of the x1 and x2 and y1 and y2

Mentor
Then maybe you're not able to work this problem...

ibaforsale
does it mean that they are on the circle?

Mentor
You should already know they're on the circle, so each of those points should satisfy that equation. Look at what you have in post #9, and use the fact that the points are on the circle.

ibaforsale
is it that the 2 points on the circle cancel each other out?