# Vector Triple products

#### Prove It

##### Well-known member
MHB Math Helper
Hi everyone,

This semester I was asked to lecture Calculus 2 at the university in which I work. I gladly accepted

Anyway, we are up to the second module, which is Vectors. Today's lecture was about Scalar Triple Products and Vector Triple Products. My class is very attentive and was asking a lot of questions today. Unfortunately, the lecture notes I inherited had ZERO information on why we would need to evaluate a vector triple product, though it said there are numerous applications including mathematical modelling of particle and fluid dynamics. Scalar triple products are easy - we use them to evaluate the volume of a paralleliped and to determine if vectors or points are coplanar. Among the participation from the students, I was asked why we would need to evaluate the vector triple product, and for the life of me I could not think of a single application. I even checked Google and could not find one there either, although geometrically we can use the vector triple product to find a vector that lies in the same plane as the final two vectors.

Just to be clear, I'm talking about $$\displaystyle \mathbf{a} \times \left( \mathbf{b} \times \mathbf{c} \right)$$, which can be evaluated more easily using $$\displaystyle \left( \mathbf{a} \cdot \mathbf{c} \right) \mathbf{b} - \left( \mathbf{a} \cdot \mathbf{b} \right) \mathbf{c}$$.

So my question is, could somebody please give me some real-world examples of applications of the vector triple product? Thanks

Staff member

#### Prove It

##### Well-known member
MHB Math Helper
It shows up in rotating reference frames, where you have to write down Newton's Second Law quite a bit differently.
Thank you Ackbach, at least I now have something to tell my students next lesson

#### Ackbach

##### Indicium Physicus
Staff member
Thank you Ackbach, at least I now have something to tell my students next lesson
You're very welcome!

#### Klaas van Aarsen

##### MHB Seeker
Staff member
The triple vector product is the volume of the body it spans.
Working it out geometrically also shows you why the identity holds.

#### Ackbach

##### Indicium Physicus
Staff member
The triple vector product is the volume of the body it spans.
Working it out geometrically also shows you why the identity holds.
Are you sure you're not thinking about the scalar triple product?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Are you sure you're not thinking about the scalar triple product?
Oops. You're right.

#### Plato

##### Well-known member
MHB Math Helper
Anyway, we are up to the second module, which is Vectors. Today's lecture was about Scalar Triple Products and Vector Triple Products. My class is very attentive and was asking a lot of questions today. Unfortunately, the lecture notes I inherited had ZERO information on why we would need to evaluate a vector triple product, though it said there are numerous applications including mathematical modelling of particle and fluid dynamics. Scalar triple products are easy - we use them to evaluate the volume of a paralleliped and to determine if vectors or points are coplanar. Among the participation from the students, I was asked why we would need to evaluate the vector triple product, and for the life of me I could not think of a single application. I even checked Google and could not find one there either, although geometrically we can use the vector triple product to find a vector that lies in the same plane as the final two vectors.

Just to be clear, I'm talking about $$\displaystyle \mathbf{a} \times \left( \mathbf{b} \times \mathbf{c} \right)$$, which can be evaluated more easily using $$\displaystyle \left( \mathbf{a} \cdot \mathbf{c} \right) \mathbf{b} - \left( \mathbf{a} \cdot \mathbf{b} \right) \mathbf{c}$$.

So my question is, could somebody please give me some real-world examples of applications of the vector triple product? Thanks
Since I know next to nothing about applied mathematics, I know there are other geometric uses for the triple vector product.
It is essential in the proof of the formula for the distance between two skew lines,

Also in defining the unit tangent vector Big T as $$\displaystyle T=\frac{R'}{\|R'\|}$$, we can use the triple vector product to simplify its derivative $$\displaystyle T'=\frac{R'\times(R''\times R')}{\|R'\|^3}$$.

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#### Prove It

##### Well-known member
MHB Math Helper
Since I know next to nothing about applied mathematics, I know there are other geometric uses for the triple vector product.
It is essential in the proof of the formula for the distance between two skew lines,

Also in defining the unit tangent vector Big T as $$\displaystyle T=\frac{R'}{\|R'\|}$$, we can use the triple vector product to simplify its derivative $$\displaystyle T'=\frac{R'\times(R''\times R')}{\|R'\|^3}$$.
Thank you Plato, since I'm from a more pure mathematics background, that helps a lot too

#### Klaas van Aarsen

##### MHB Seeker
Staff member
If you have 2 vectors $\mathbf a$ and $\mathbf b$, then $\{ \mathbf a, (\mathbf a \times \mathbf b), \mathbf a \times (\mathbf a \times \mathbf b) \}$ forms an orthogonal basis.

This is basically the Gramm-Schmidt orthogonalization process.

That's also what happens in Plato's example.
$\mathbf T$ is the unit velocity vector, while $\mathbf T' = \kappa \mathbf N$ is the vector perpendicular to it (curvature times unit normal vector). This is how curvature is defined as a component of a local orthonormal basis.

And that's also what happens in Ackbach's example, where basically each direction in a local orthonormal basis is labeled with a different name.