Find parametric equations for the line through the point $( 3,4,5 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$.
A) $x = 4 t + 2 , y = - 2 t + 4 , z = - 2 t + 5$
B) $x = 4 t + 2 , y = - 2 t + 4 , z = 0$
C) $x = 4 t , y = - 2 t , z = - 2 t + 5$
D) $x = 4 t , y = - 2 t + 4 , z = - 2 t$
E) $x = 4 t + 3 , y = - 2 t + 4 , z = - 2 t + 5$

Question

Quizplus · Accepted Answer

The direction vector of the line must be perpendicular to the normal vector of the plane (1,1,1) and parallel to the direction vector of the given line (1,-1,3). The cross product of (1,1,1) and (1,-1,3) gives (4,-2,-2). Using point (3,4,5), the parametric equations are $x = 4t + 3$, $y = -2t + 4$, $z = -2t + 5$.

Find Parametric Equations for the Line Through the Point (3,4,5)( 3,4,5 )(3,4,5)

Find Parametric Equations for the Line Through the Point $( 3,4,5 )$