Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

A friend.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 97,132

I will look at it today but do not expect too much.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

Offline

Solution

Let us call the curves c1, c2 and c3

It is evident that:

c1 = (cos t, sin t, 0)

c2 = (cos t, 0, sin t)

c3 = (0, sin t, cos t)

[I am considering the circles to be centered at the origin and of radius 1. This is just to make the algebra simple]

You wanted me to prove that c1 intersects c2 making an angle of pi/2 and c2 intersects c3 making an angle of pi/2 and c3 intersects c1 making an angle of pi/2

Consider the intersection of c1 and c2. At this point, c1(t) = c2(t)

Or, (cos t, sin t, 0) = (cos t, 0, sin t)

Solving, you get t = 0

Now, c1(0) = c2(0) = (1, 0, 0)

Differentiating the curves,

c1' = (-sin t, cos t, 0)

c2' = (-sin t, 0, cos t)

At t=0, c1'(0) = (0, 1, 0) and c2'(0) = (0, 0, 1)

Let x1 be the tangent of c1 at the intersection point

We have, x1= c1(0) + t(c1'(0)) = (1, 0, 0) + t(0, 1, 0) = (1, t, 0)

Let x2 be the tangent of c2 at the intersection point

We have, x2= c2(0) + t(c2'(0)) = (1, 0, 0) + t(0, 0, 1) = (1, 0, t)

By the angle between c1 and c2, we obviously mean the angle between x1 and x2.

Let that angle be θ

x1 . x2 = (1, t, 0) . (1, 0, t) = 1*1 + t*0 + t*0 = 1

|x1| = Sqrt[1+t^2]

|x2| = Sqrt[1+t^2]

At the intersection point, t=0. So,

Similarly, you can go on proving the same thing for the two other intersection points (but that has been left to the reader as an exercise )

QED

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Offline