Let P be the plane such that it contains the straight line $\frac{x-1}{2}=\frac{y-3}{3}=\frac{z+2}{1}$ and is perpendicular to the plane $x+2y+3z=4$. Let $P_{1}$ be the plane which passes through the point $(4,2,2)$ and is parallel to P. Then which of the following statements is (are) TRUE?

$\text{Given: } P \supset \dfrac{x-1}{2} = \dfrac{y-3}{3} = \dfrac{z+2}{1}$

$P \perp x + 2y + 3z = 4$

$P_1 \parallel P, \ P_1 \ni (4,2,2)$

$\text{To find: which of the statements (1)--(4) are TRUE}$

$1.\ \text{find } \vec{n}_P$

$2.\ \text{equation of } P$

$3.\ \text{test each statement}$

$\vec{d} = (2, 3, 1), \quad A(1, 3, -2) \in L$

$L \subset P \Rightarrow \vec{n}_P \cdot \vec{d} = 0$

$P \perp (x+2y+3z=4) \Rightarrow \vec{n}_P \cdot (1,2,3) = 0$

$\vec{n}_P = \vec{d} \times (1,2,3) = (7, -5, 1)$

$7(x-1) - 5(y-3) + (z+2) = 0$

$P:\ 7x - 5y + z = -10 \Rightarrow \text{(1) TRUE}$

$P_1:\ 7x - 5y + z = 28 - 10 + 2 = 20$

$d(P, P_1) = \dfrac{|20 - (-10)|}{\sqrt{75}} = \dfrac{30}{5\sqrt3} = 2\sqrt3$

$\Rightarrow \text{(2) false (says 30)}$

$d_O = \dfrac{|10|}{\sqrt{75}} = \dfrac{2}{\sqrt3} = \dfrac{2\sqrt3}{3}$

$\Rightarrow \text{(3) false}$

$\cos\theta = \dfrac{|(7,-5,1)\cdot(2,2,1)|}{\sqrt{75}\cdot 3} = \dfrac{5}{15\sqrt3} = \dfrac{1}{3\sqrt3}$

$\Rightarrow \text{(4) TRUE}$

$\text{True: } (1)\ \text{and}\ (4)$

$\boxed{(1)\ \text{and}\ (4)}$