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 don't see how those two points along with a third make a right triangle.

 

[Mark44]

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 don't 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)?

 

[Mark44]

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, let's 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?

 

[LCKurtz]

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)

 

[LCKurtz]

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 that's not 0

 

[Mark44]

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 that's 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

 

[Mark44]

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 I am not sure if that would work in the general sense.

 

[Mark44]

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

 

[Mark44]

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) isn't because when you plug it into the equation it doesn't 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?

 

[Mark44]

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, I am a little thrown off because of the x1 and x2 and y1 and y2

 

[Mark44]

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

 

[ibaforsale]

does it mean that they are on the circle?

 

[Mark44]

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?

 

[LCKurtz]

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

Write an equation so we know what you are trying to say.

 

[Mark44]

Write an equation so we know what you are trying to say.

+1 to that.