n-dimensional trianglehedron?

[T@P]

if you take two unit orthogonal vectors in R2, the triangle they form has area 1/2. take 3 in R3, the volume is 1/6. i claim (and half proved) that the voluarea of an n dimensional bunch of orthognal vectors would give you 1/n!. can anyone prove it fully?

(go easy on the linear algebra, im infantile when it comes to that)

[NateTG]

I take it you mean the n-volume of the convex hull formed by the origin and the usual unit vectors in \Re^n under the usual metric.

Let's instead of looking at just the unit length version, look at v_n(r) the 'trianglehedron' with legs of length r.

Now, let's prove by induction that
v_n(r)=\frac{1}{n!} r^n
Then it's easy to see that
v_1(r)=\frac{1}{n!} r^1
And that for the n+1 case, we can integrate by slicing to get:
v_{n+1}(r)=\int_0^rv_n(x)dx
but from induction we have:
v_n(x)=\frac{1}{n!}x^n
so
v_{n+1}(r)=\int_0^r\frac{1}{n!}x^n dx
which readily works out to
v_{n+1}(r)=\frac{1}{n!} \times \frac{1}{n+1} r^{n+1}=\frac{1}{(n+1)!}r^{n+1}

Sepecifically,
v_n(1)=\frac{1}{n!}