In space, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$. Let $\mathscr { D } _ { 2 }$ be the line with parametric representation $\left\{ \begin{aligned} x & = 1 + t \\ y & = - 3 - t \\ z & = 2 - 2 t \end{aligned} \quad ( t \in \mathbb { R } ) \right.$. a. The line $\mathscr { D } _ { 2 }$ and the plane $\mathscr { P }$ are not secant. b. The line $\mathscr { D } _ { 2 }$ is contained in the plane $\mathscr { P }$. c. The line $\mathscr { D } _ { 2 }$ and the plane $\mathscr { P }$ intersect at the point $\mathrm { E } \left( \frac { 1 } { 3 } ; - \frac { 7 } { 3 } ; \frac { 10 } { 3 } \right)$. d. The line $\mathscr { D } _ { 2 }$ and the plane $\mathscr { P }$ intersect at the point $\mathrm { F } \left( \frac { 4 } { 3 } ; - \frac { 1 } { 3 } ; \frac { 22 } { 3 } \right)$.
In space, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$.
Let $\mathscr { D } _ { 2 }$ be the line with parametric representation $\left\{ \begin{aligned} x & = 1 + t \\ y & = - 3 - t \\ z & = 2 - 2 t \end{aligned} \quad ( t \in \mathbb { R } ) \right.$.\\
a. The line $\mathscr { D } _ { 2 }$ and the plane $\mathscr { P }$ are not secant.\\
b. The line $\mathscr { D } _ { 2 }$ is contained in the plane $\mathscr { P }$.\\
c. The line $\mathscr { D } _ { 2 }$ and the plane $\mathscr { P }$ intersect at the point $\mathrm { E } \left( \frac { 1 } { 3 } ; - \frac { 7 } { 3 } ; \frac { 10 } { 3 } \right)$.\\
d. The line $\mathscr { D } _ { 2 }$ and the plane $\mathscr { P }$ intersect at the point $\mathrm { F } \left( \frac { 4 } { 3 } ; - \frac { 1 } { 3 } ; \frac { 22 } { 3 } \right)$.