Re:inner angle of a triangle in the 4th dimension

• To: mathgroup at smc.vnet.net
• Subject: [mg26335] Re:[mg26324] inner angle of a triangle in the 4th dimension
• From: "Ingolf Dahl" <f9aid at fy.chalmers.se>
• Date: Tue, 12 Dec 2000 02:54:47 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi!
Be a bit careful with signs:
Define w as RP -> {-3, -4, 1, -2}, so that you walk around the triangle in a
systematic way. It is easier to get the signs right then.

Then u = {3, 1, -2, 0}; v = {0, 3, 1, 2}; w = {-3, -4, 1, -2};

(You can check that u + v + w == 0)

The length of u is Sqrt[u.u]=Sqrt[14]
The length of v is Sqrt[v.v]=Sqrt[14]
The length of w is Sqrt[w.w]=Sqrt[30]

You should not put ABS inside the ArcCos calculation. You have to be careful
about the signs of the vectors, otherwise you calculate the outer angle
instead of the inner angle for some triangle corners, in an unsystematic
way.

The inner angle at Q:
N[ArcCos[(-u).v/(Sqrt[u.u]*Sqrt[v.v])]/Degree] = 94.096 deg

The inner angle at R:
N[ArcCos[(-v).w/(Sqrt[v.v]*Sqrt[w.w])]/Degree] = 42.952 deg

The inner angle at P:
N[ArcCos[(-w).u/(Sqrt[w.w]*Sqrt[u.u])]/Degree] = 42.952 deg

The sum is 180 deg.

Ingolf Dahl
Chalmers University

> Jacky Vaillancourt wrote in [mg26324]:
Hi, i have a basic problem. I can't see my mistake can somebody help me?

Here's the problem:
I want to calculate each angle of the triangle formed by those three dots.
P:=(0,1,0,1), Q:=(3,2,-2,1), R:=(3,5,-1,3)

u:=PQ -> (3-0,2-1,-2-0,1-1) -> (3,1,-2,0)
v:=QR -> (3-3,5-2,-1-(-2),3-1) -> (0,3,1,2)
w:=PR -> (3-0,5-1,-1-0,3-1) -> (3,4,-1,2)

The formula to have the angle between tho vector is:
ARCCOS(ABS(DOTPROD(u,v))/(length(u)*length(v))

The formula to calculate the length is SQRT(a^2+b^2+c^2+d^2)

So, the angle between u and v is:
ARCCOS(ABS(15)/(SQRT(14)*SQRT(30))) = 42.95 deg

the angle between v and w is:
ARCCOS(ABS(-15)/(SQRT(30)*SQRT(14)))= 42.95 deg

the angle between u and w is:
ARCCOS(ABS(-1)/(SQRT(30)*SQRT(14)))= 85.9 deg

Here's the problem 180-85.9-42.95-42.95= 8.2 deg

I'm missing 8.2 deg....

I hope you'll understand what i wrote, i'm not used to write in english...

Thanks

Jacky

```

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