What is the Maximum Value of Lambda in this Vector Algebra Challenge?

[Saitama]

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three unit vectors such that $\left|\vec{a}+\vec{b}+\vec{c}\right|=\sqrt{3}$ and $\left(\vec{a}\times\vec{b}\right)\cdot \left(\vec{b}\times\vec{c}\right)+\left(\vec{b}\times\vec{c}\right)\cdot \left(\vec{c}\times\vec{a}\right)+\left(\vec{c}\times\vec{a}\right)\cdot \left(\vec{a}\times\vec{b}\right)=\lambda$.

i) Find the maximum value of $\lambda$.

ii) Corresponding to this maximum value of $\lambda$, find

a)The volume of parallelepiped determined by $\vec{a}$, $\vec{b}$ and $\vec{c}$ is $\sqrt{3}$.

b)The value of $\left|\left(2\vec{a}+3\vec{b}+4\vec{c}\right)\cdot \left(\vec{a}\times \vec{b}+5\vec{b}\times \vec{c}+6\vec{c}\times \vec{a}\right)\right|$.

 

[jvicens]

Thank you for your interesting question. I would like to provide you with the answers to your questions.

i) To find the maximum value of $\lambda$, we can use the Cauchy-Schwarz inequality, which states that for any two vectors $\vec{u}$ and $\vec{v}$, we have $\left|\vec{u}\cdot\vec{v}\right|\leq \left|\vec{u}\right|\cdot\left|\vec{v}\right|$. Applying this to the given equation, we have
$$\left|\left(\vec{a}\times\vec{b}\right)\cdot \left(\vec{b}\times\vec{c}\right)+\left(\vec{b}\times\vec{c}\right)\cdot \left(\vec{c}\times\vec{a}\right)+\left(\vec{c}\times\vec{a}\right)\cdot \left(\vec{a}\times\vec{b}\right)\right|\leq \left|\vec{a}\times\vec{b}\right|\cdot\left|\vec{b}\times\vec{c}\right|+\left|\vec{b}\times\vec{c}\right|\cdot\left|\vec{c}\times\vec{a}\right|+\left|\vec{c}\times\vec{a}\right|\cdot\left|\vec{a}\times\vec{b}\right|$$
$$=\left|\vec{a}\times\vec{b}\right|\cdot\left|\vec{b}\right|\cdot\left|\vec{c}\right|+\left|\vec{b}\times\vec{c}\right|\cdot\left|\vec{c}\right|\cdot\left|\vec{a}\right|+\left|\vec{c}\times\vec{a}\right|\cdot\left|\vec{a}\right|\cdot\left|\vec{b}\right|=\left|\vec{a}\right|\cdot\left|\vec{b}\right|\cdot\left|\vec{c}\right|\cdot\left(\left|\vec{a}\right|+\left|\vec{b}\right|+\left|\vec{c}\right|\right)=\left|\vec{a}+\vec{b}+\vec{c}\right