Let $\vec{v}=\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times ....$

Question

Problem :

Let $\vec{v}=\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times (\vec{j}\times (\vec{j} \times \vec{k}))))))))$ Then find the value of $||\vec{v}||$

I am not getting any idea how to proceed in this, please suggest , will be of great help. Thanks.

Answer 1

Work from inside to out, starting with $j \times k=i,$ at first glance it seems you'll end up at $\pm$ one of $i,j,k$ so norm is $1.$

Added: The quaternions are a division ring, in particular are associative. Since $ijk=ii=-1,$ the first two of $ijkijkijjk$ make $+1,$ and then $ijjk=-ik=j.$

Answer 2

As always, the cross product supplies you with a vector that is orthogonal to both vectors, with a magnitude equal to the parallelogram made by them.

Crossing any of the three unit vectors will give a new positive or negative $i, j, k$. As such, no matter how many you cross, you will always have a unit vector. This has a magnitude of $1$.

Answer 3

Hint:

$$\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times (\vec{j}\times (\vec{j} \times \vec{k}))))))))$$

$$\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times (\vec{j}\times \vec{i})))))))$$

$$-\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times\vec{k}))))))$$

$$\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times (\vec{j}\times (\vec{k}\times \vec{j})))))$$

$$-\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times (\vec{j}\times \vec{i}))))$$

$$\vec{i}\times (\vec{j}\times (\vec{k}\times (\vec{i}\times \vec{k})))$$

$$-\vec{i}\times (\vec{j}\times (\vec{k}\times \vec{j}))$$

$$\vec{i}\times (\vec{j}\times \vec i)$$

$$-\vec{i}\times \vec k$$

$$\vec{j}$$

You must check all intermediate results.

Answer 4

You just have to be methodical about it.

$$ \begin{align} \vec{v}&=\vec{i}\times(\vec{j}\times(\vec{k}\times(\vec{i}\times(\vec{j}\times(\vec{k}\times(\vec{i}\times(\vec{j}\times(\vec{j}\times\vec{k}))))))))&&\vec{j}\times\vec{k}=\vec{i}\\&=\vec{i}\times(\vec{j}\times(\vec{k}\times(\vec{i}\times(\vec{j}\times(\vec{k}\times(\vec{i}\times(\vec{j}\times\vec{i})))))))&&\vec{j}\times\vec{i}=-\vec{k}\\&=\vec{i}\times(\vec{j}\times(\vec{k}\times(\vec{i}\times(\vec{j}\times(\vec{k}\times(\vec{i}\times(-\vec{k})))))))&&\vec{i}\times\left(-\vec{k}\right)=\vec{j}\\&=\vec{i}\times(\vec{j}\times(\vec{k}\times(\vec{i}\times(\vec{j}\times(\vec{k}\times\vec{j})))))&&\vec{k}\times\vec{j}=-\vec{i}\\&=\vec{i}\times(\vec{j}\times(\vec{k}\times(\vec{i}\times(\vec{j}\times(-\vec{i})))))&&\vec{j}\times\left(-\vec{i}\right)=\vec{k}\\&\vdots&& \end{align} $$