Statement -1 : The cartesian equation of the plane $\vec{r} = (\hat{i} - \hat{j}) + λ(\hat{i} + \hat{j} + \hat{k}) + \mu (\hat{i} -2\hat{j} + 3\hat{k})$ is 5x - 2y - 3z = 7.

Statement-2: The non-parametric form of the plane $\vec{r} = \vec{a} + λ \vec{b} + \mu \vec{c}$ is $[\vec{r} \, \vec{b}\, \vec{c}]= [\vec{a} \, \vec{b}\, \vec{c}]$

Statement 1 is True, Statement 2 is true; Statement 2 is a correct explanation for Statement 1.

Equation $\vec{r} = \vec{a} + λ \vec{b} + \mu \vec{c}$ represents a plane passing through point $\vec{a}$ and parallel to vectors $\vec{b}$ and $\vec{c}$. So, it is normal to vector $\vec{n} = \vec{b}× \vec{c}$. Therefore, equation of the plane in non-parametric form can be written as

$(\vec{r} - \vec{a}). (\vec{b}× \vec{c})= 0 $

$⇒\vec{r}.(\vec{b}× \vec{c})= \vec{a}.(\vec{b}× \vec{c})⇒[\vec{r} \, \vec{b}\, \vec{c}]= [\vec{a} \, \vec{b}\, \vec{c}]$

So, statement-2 is true.

For the given plane $\vec{r} = (\hat{i} - \hat{j}) + λ(\hat{i} + \hat{j} + \hat{k}) + \mu (\hat{i} -2\hat{j} + 3\hat{k})$

We have $\vec{r} = \hat{i} - \hat{j}, \vec{b}= \hat{i} + \hat{j} + \hat{k}$ and $ \vec{c}= \hat{i} - 2\hat{j} +3\hat{k}.$

So, its cartesian equations is

$\begin{bmatrix}x & y & z\\1 & 1 & 1\\1 & -2 & 3\end{bmatrix}=\begin{bmatrix}1 & -1 & 0\\1 & 1 & 1\\1 & -2 & 3\end{bmatrix}= 5x - 2y - 3z = 7 $

So, statement-1 is true and statement-2 is a correct explanation for statement-1.