Statement 1: The vectors $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ lie in the same plane if and only if $$\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$$. Statement 2: The vectors $$\vec{u}$$ and $$\vec{v}$$ are perpendicular if and only if $$\vec{u} \cdot \vec{v} = 0$$ where $$\vec{u} \times \vec{v}$$ is a vector perpendicular to the plane of $$\vec{u}$$ and $$\vec{v}$$.

Let $$\vec{a},\vec{b},\vec{c}\in\mathbb{R}^{3}$$ and recall two standard results:

1. Scalar triple product: $$[\vec{a}\;\vec{b}\;\vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c})$$ equals the volume of the parallelepiped formed by the three vectors (with sign).
2. Dot product: $$\vec{u}\cdot\vec{v}=|\vec{u}\,||\vec{v}|\cos\theta$$ where $$\theta$$ is the angle between $$\vec{u}$$ and $$\vec{v}$$.

Statement 1: “The vectors $$\vec{a},\vec{b},\vec{c}$$ lie in the same plane (are coplanar) $$\Leftrightarrow$$ $$\vec{a}\cdot(\vec{b}\times\vec{c})=0$$.”

If $$\vec{a},\vec{b},\vec{c}$$ are coplanar, the parallelepiped collapses into a parallelogram, so its volume is zero; hence their scalar triple product is zero.
Conversely, if $$\vec{a}\cdot(\vec{b}\times\vec{c})=0$$, the volume is zero, so the three vectors cannot span all of $$\mathbb{R}^{3}$$; they must lie in the same plane through the origin. Thus Statement 1 is true.

Statement 2: “The vectors $$\vec{u}$$ and $$\vec{v}$$ are perpendicular if and only if $$\vec{u}\cdot\vec{v}=0$$ (where $$\vec{u}\times\vec{v}$$ is a vector perpendicular to the plane of $$\vec{u},\vec{v}$$).”

The familiar condition $$\vec{u}\cdot\vec{v}=0$$ indeed implies $$\theta=90^{\circ}$$ provided both $$\vec{u}$$ and $$\vec{v}$$ are non-zero. The statement, however, omits this necessary non-zero requirement. For example, choose

$$\vec{u}=\vec{0},\qquad \vec{v}=(1,0,0).$$

Then $$\vec{u}\cdot\vec{v}=0$$, yet it is meaningless to call the zero vector perpendicular to any vector because it has no direction. Thus “$$\vec{u}\cdot\vec{v}=0$$” is not equivalent to perpendicularity unless we separately assume $$\vec{u}\neq\vec{0}$$ and $$\vec{v}\neq\vec{0}$$. Therefore Statement 2 is false.

Since Statement 1 is true and Statement 2 is false, Statement 2 cannot explain Statement 1. Hence the correct option is:

Option C which is: Statement 1 is true, Statement 2 is false.