Consider the following statements in respect of a vector d = ( a ×...

Given $\vec{d} = (\vec{a} \times \vec{b}) \times \vec{c}$

Using the vector triple product expansion:

$\vec{d} = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{b} \cdot \vec{c})\vec{a}$

This shows that $\vec{d}$ is expressed as a linear combination of $\vec{a}$ and $\vec{b}$. Thus, $\vec{d}$ lies in the plane containing $\vec{a}$ and $\vec{b}$, making it coplanar with $\vec{a}$ and $\vec{b}$. Statement I is correct.

By the definition of the cross product, the vector resulting from the cross product of two vectors is perpendicular to both of them. Since $\vec{d}$ is the cross product of $(\vec{a} \times \vec{b})$ and $\vec{c}$, it must be perpendicular to both $(\vec{a} \times \vec{b})$ and $\vec{c}$.

Therefore, $\vec{d} \cdot \vec{c} = 0$, which means $\vec{d}$ is perpendicular to $\vec{c}$. Statement II is correct.

Both statements I and II are correct.

Answer: Both I and II