Vectors: Find w such that dv/ds = w x v

1. The problem statement, all variables and given/known data

T (tangent), B (binormal), N (normal) are an orthogonal triad of unit vectors of a curve in R3.


Given:

• dT/ds = kN
• dN/ds = -kT + tB
• dB/ds = -tN

Find vector w so that these equations may be written in the form:

dv/ds = w x v, where v = T + N + B




2. Relevant equations

Given above!




3. The attempt at a solution

I tried splitting v (T + N + B) into its components, as well as w, and putting them into a matrix to find the determinant (for the cross product).

The matrix consists of <i, j, k>, w = <w1, w2, w3>, and v = <(T1 + N1 + B1), (T2 + N2 + B2), (T3 + N3 + B3)>.

However, taking the determinant/cross product gives mismatching components (ie. j and k components supposedly add up to an i component). I'm not too sure where to go from here :(



Note: This is for a Multivariate Calculus course!

 

First of all, thanks for the welcome! :)

I know the hint you gave me means [itex]\left(\mathbf{v} \bullet \mathbf{v}\right)\mathbf{w} - \left(\mathbf{v} \bullet \mathbf{w}\right)\mathbf{v}[/itex]. I've tried this method before, but I wasn't too sure where to go from here..