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

• May 15th 2012, 09:54 AM
x3bnm
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:

$b = n \times t$

where $b =$ binormal and $n$ is normal to the curve and $t$ is tangent.

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

Now if $b = n \times t$ why's that $n = b \times t$ ? How do one prove this statement?

What properties of mathematics verify that $n = b \times t$? Is it possible to kindly help me answer this question?
• May 15th 2012, 10:30 AM
HallsofIvy
Re: Why's that normal = binormal x tangent for a curve parametrized by arc length s?
With $b= n\times t$, there is NO "property of mathematics" that gives " $n= b\times t$". It is, rather, true that $n= t\times b$.

n and t are also unit vectors. Further, the length of $u\times v$ is $|u||v|sin(\theta)$ where $\theta$ 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, $b= n\times t$, $n= t\times b$ and $t= b\times n$.
• May 15th 2012, 10:56 AM
x3bnm
Re: Why's that normal = binormal x tangent for a curve parametrized by arc length s?
Thanks HallsofIvy.

Then if $a = b \times c$ for any vector $a, b$ and $c$ then can we say $b = a \times c$ and $c = a \times b$?
• May 15th 2012, 11:45 AM
x3bnm
Re: Why's that normal = binormal x tangent for a curve parametrized by arc length s?
Quote:

x3bnm said:
Then if $a = b \times c$ for any vector $a, b$ and $c$ then can we say $b = a \times c \text{ and } c = a \times b$?

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

Suppose $b = (1,2,1), c = (2, 3, 1) \text{ and } a = b \times c = (-1, 1, -1)$

But $b \neq a \times c \text{ and } c \neq a \times b$

The statement that $n = b \times t$ is true for binormal, normal and tangent.