With a laser device located at point $\mathrm { P } ( 1,1,1 )$ it has been possible to follow the trajectory of a particle that moves along the line with equations $r \equiv \left\{ \begin{array} { l } 2 x - y = 10 \\ x - z = - 90 \end{array} \right.$. a) ( 0.5 points) Calculate a direction vector of $r$ and the position of the particle when its trajectory intersects the plane $z = 0$. b) (1.25 points) Calculate the closest position of the particle to the laser device. c) ( 0.75 points) Determine the angle between the plane with equation $x + y = 2$ and the line $r$.
With a laser device located at point $\mathrm { P } ( 1,1,1 )$ it has been possible to follow the trajectory of a particle that moves along the line with equations $r \equiv \left\{ \begin{array} { l } 2 x - y = 10 \\ x - z = - 90 \end{array} \right.$.\\
a) ( 0.5 points) Calculate a direction vector of $r$ and the position of the particle when its trajectory intersects the plane $z = 0$.\\
b) (1.25 points) Calculate the closest position of the particle to the laser device.\\
c) ( 0.75 points) Determine the angle between the plane with equation $x + y = 2$ and the line $r$.