Why's that normal = binormal x tangent for a curve parametrized by arc length s?

I'm stuck at the properties of a 3d curve parametrized by arc length. My problem is given below.

We know that the binormal of a curve at a point is:

where binormal and is normal to the curve and is tangent.

So binormal is the cross product of normal and tangent at a point of a curve.

Now if why's that ? How do one prove this statement?

What properties of mathematics verify that ? Is it possible to kindly help me answer this question?

Re: Why's that normal = binormal x tangent for a curve parametrized by arc length s?

With , there is NO "property of mathematics" that gives " ". It is, rather, true that .

n and t are also **unit** vectors. Further, the length of is where is the angle between the two vectors. Since n and t are at right angles, b is also a unit vector, perpendicular to both n and t. That is, we have the "cyclic" formulas, , and .

Re: Why's that normal = binormal x tangent for a curve parametrized by arc length s?

Thanks HallsofIvy.

Then if for any vector and then can we say and ?

Re: Why's that normal = binormal x tangent for a curve parametrized by arc length s?

Quote:

x3bnm said:

Then if

for any vector

and

then can we say

?

I got it. My last statement is wrong. For those who want to understand: the Counterexample to my statement:

Suppose

But

The statement that is true for binormal, normal and tangent.