Vector Triple Product - Physcial Significance

[RanchodDasChanchad]

Hii,

As we know, Scaler triple product is volume of parallelopiped constructed by its three sides.

Similary,

What is the physical significance and geometrical interpretation of Vector triple product ?

Also, What are the application where we use such mathematics and why ?

Regards,
Rahul

 

[Orodruin]

A good question. Since the cross product $\vec v \times \vec w$ is a vector orthogonal to both $\vec v$ and $\vec w$, the product $\vec v \times (\vec w \times \vec u)$ is a vector that is orthogonal to both $\vec v$ and $\vec w \times \vec u$. As such, it lies in the plane spanned by $\vec w$ and $\vec u$ (since $\vec w$ and $\vec u$ are orthogonal to $\vec w \times \vec u$). Now, this becomes particularly interesting when $\vec u = \vec v$, since the result must be then be a linear combination of $\vec v$ and $\vec w$ at the same time as it is orthogonal to $\vec v$, which leaves only one possible direction. Let us see what use we can put this to.

Consider the case when $\vec v = \vec n$ is a normalised vector of length one, i.e., $\vec n^2 = 1$ and take the triple product
$$
\vec n \times (\vec w \times \vec n) = \vec w (\vec n^2) - (\vec n \cdot \vec w) \vec n = \vec w - (\vec n \cdot \vec w) \vec n,
$$
where we have applied the bac-cab rule. Rearranging this leads to the relation
$$
\vec w = \vec n (\vec w \cdot \vec n) + \vec n \times (\vec w \times \vec n).
$$
You may recognise the first term on the right as the projection of $\vec w$ on $\vec n$. As both terms add up to $\vec w$, it should therefore be clear that $\vec n \times (\vec w \times \vec n)$ is the component of $\vec w$ orthogonal to $\vec n$. One common use of the vector triple product is therefore to find the orthogonal component of a vector with respect to another.

 

[Orodruin]

Ah yes, we can also add that the particular linear combination of $\vec b$ and $\vec c$ that $\vec a \times (\vec b \times \vec c)$ is given by can be found through the bac-cab-rule
$$
\vec a \times (\vec b \times \vec c) = \vec b (\vec a \cdot \vec c) - \vec c (\vec a \cdot \vec b).
$$
Clearly this is a vector that is in the plane spanned by $\vec b$ and $\vec c$. Furthermore, the orthogonality to $\vec a$ is apparent (dotting $\vec a$ into the expression gives two terms with the same size and opposite signs).