Let $L_1$ be the line of intersection of the planes given by the equations $2x + 3y + z = 4$ and $x + 2y + z = 5$. Let $L_2$ be the line passing through the point $P(2,-1,3)$ and parallel to $L_1$. Let $M$ denote the plane given by the equation $2x + y - 2z = 6$. Suppose that the line $L_2$ meets the plane $M$ at the point $Q$. Let $R$ be the foot of the perpendicular drawn from $P$ to the plane $M$. Then which of the following statements is (are) TRUE?

Let L₁ be the line of intersection of the planes given by the equations 2x + 3y + z = 4 and x + 2y + z = 5. Let L₂ be the line passing through the point P(2,-1,3) and parallel to L₁. Let M denote the plane given by the equation 2x + y - 2z = 6. Suppose that the line L₂ meets the plane M at the point Q. Let R be the foot of the perpendicular drawn from P to the plane M. Then which of the following statements is (are) TRUE? $L_1 = \pi_1 \cap \pi_2$ $\pi_1: 2x+3y+z=4,\ \ \pi_2: x+2y+z=5$ $L_2 \parallel L_1 \text{ through } P(2,-1,3),\ \ M: 2x+y-2z=6$ $Q = L_2 \cap M,\quad R = \text{foot of } \perp \text{ from } P \text{ to } M$ $\text{Check: } PQ,\ QR,\ [\triangle PQR],\ \angle QPR$ $\text{Plan: } 1.\ \vec{d} = \vec{n_1}\times\vec{n_2} \Rightarrow L_2$ $2.\ Q = L_2 \cap M$ $3.\ R = \text{foot of } \perp \text{ from } P;\ \text{then } PQ,\ QR,\ [\triangle PQR],\ \angle QPR$ $\vec{n_1}=(2,3,1),\quad \vec{n_2}=(1,2,1)$ $\vec{d}=\vec{n_1}\times\vec{n_2}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\2&3&1\\1&2&1\end{vmatrix}=(1,-1,1)$ $L_2:\ \dfrac{x-2}{1}=\dfrac{y+1}{-1}=\dfrac{z-3}{1}$ $L_2: (x,y,z)=(2,-1,3)+t(1,-1,1)$ $x=2+t,\ \ y=-1-t,\ \ z=3+t$ $2(2+t)+(-1-t)-2(3+t)=6$ $-3-t=6 \Rightarrow t=-9$ $Q=(-7,8,-6)$ $\vec{PQ}=Q-P=(-9,9,-9)$ $|PQ|=\sqrt{81+81+81}=9\sqrt{3}$ $\text{Statement (1): TRUE}$ $PR=\dfrac{|2(2)+(-1)-2(3)-6|}{\sqrt{2^2+1^2+2^2}}=\dfrac{9}{3}=3$ $\vec{PR}=R-P=(2,1,-2)$ $R=(4,0,1)$ $\vec{QR}=R-Q=(11,-8,7)$ $|QR|=\sqrt{121+64+49}=\sqrt{234}\approx 15.3$ $\text{Statement (2): FALSE}$ $\cos\theta=\dfrac{\vec{PQ}\cdot\vec{PR}}{|PQ|\,|PR|}=\dfrac{9}{27\sqrt{3}}=\dfrac{1}{3\sqrt{3}}$ $\text{Statement (4): FALSE}$ $\vec{PQ}\times\vec{PR}=(-9,9,-9)\times(2,1,-2)=(-9,-36,-27)$ $|\vec{PQ}\times\vec{PR}|=3\sqrt{234} \Rightarrow [\triangle PQR]=\dfrac{3}{2}\sqrt{234}$ $\text{Statement (3): TRUE}$ $PQ=9\sqrt{3}\ \text{ and }\ [\triangle PQR]=\dfrac{3}{2}\sqrt{234}$ $\boxed{(1)\ \text{and}\ (3)}$