• Support PF! Buy your school textbooks, materials and every day products Here!

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

  • Thread starter YayMathYay
  • Start date
  • #1
22
0

Homework Statement



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




Homework Equations



Given above!




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!
 
Last edited:

Answers and Replies

  • #2
gabbagabbahey
Homework Helper
Gold Member
5,002
6

Homework Statement



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




Homework Equations



Given above!




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!
Hi YayMathYay, welcome to PF! :smile:

There's a much easier way to do this than by looking at the Cartesian components of the vectors.

Hint: What is [itex]\mathbf{v}\times ( \mathbf{w} \times \mathbf{v} )[/itex]?
 
  • #3
22
0
Hi YayMathYay, welcome to PF! :smile:

There's a much easier way to do this than by looking at the Cartesian components of the vectors.

Hint: What is [itex]\mathbf{v}\times ( \mathbf{w} \times \mathbf{v} )[/itex]?
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..
 
  • #4
gabbagabbahey
Homework Helper
Gold Member
5,002
6
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..
For starters, what is [itex]\mathbf{v} \cdot \mathbf{v}[/itex] ? Next since [itex]\mathbf{T}[/itex], [itex]\mathbf{B}[/itex] & [itex]\mathbf{N}[/itex] are an orthogonal triad in [itex]\mathbb{R}^3[/itex], they span [itex]\mathbb{R}^3[/itex], and so any vector can be decomposed into a linear combination of those unit vectors. So, why not write [itex]\mathbf{w} = w_{T}\mathbf{T} + w_{N}\mathbf{N} + w_{B}\mathbf{B}[/itex]... what does that make [itex]\mathbf{w} ( \mathbf{v} \cdot \mathbf{v} ) - \mathbf{v} ( \mathbf{w} \cdot \mathbf{v} )[/itex]?
 

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

  • Last Post
Replies
1
Views
998
Replies
3
Views
603
Replies
3
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
0
Views
1K
Replies
5
Views
2K
Replies
5
Views
23K
Top