Let the plane $\pi \equiv x + y + z = 1$, the line $r _ { 1 } \equiv \left\{ \begin{array} { l } x = 1 + \lambda \\ y = 1 - \lambda \\ z = - 1 \end{array} , \lambda \in \mathbb { R } \right.$ and the point $P ( 0,1,0 )$. a) ( 0.5 points) Verify that the line $r _ { 1 }$ is contained in the plane $\pi$ and that the point P belongs to the same plane. b) ( 0.75 points) Find an equation of the line contained in the plane $\pi$ that passes through P and is perpendicular to $r _ { 1 }$. c) (1.25 points) Calculate an equation of the line, $r _ { 2 }$, that passes through P and is parallel to $r _ { 1 }$. Find the area of a square that has two of its sides on the lines $r _ { 1 }$ and $r _ { 2 }$.
Let the plane $\pi \equiv x + y + z = 1$, the line $r _ { 1 } \equiv \left\{ \begin{array} { l } x = 1 + \lambda \\ y = 1 - \lambda \\ z = - 1 \end{array} , \lambda \in \mathbb { R } \right.$ and the point $P ( 0,1,0 )$.\\
a) ( 0.5 points) Verify that the line $r _ { 1 }$ is contained in the plane $\pi$ and that the point P belongs to the same plane.\\
b) ( 0.75 points) Find an equation of the line contained in the plane $\pi$ that passes through P and is perpendicular to $r _ { 1 }$.\\
c) (1.25 points) Calculate an equation of the line, $r _ { 2 }$, that passes through P and is parallel to $r _ { 1 }$. Find the area of a square that has two of its sides on the lines $r _ { 1 }$ and $r _ { 2 }$.