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 both the normal vector of the plane and the direction vector of the given line. The normal vector of the plane $x + y + z = -15$ is $(1, 1, 1)$, and the direction vector of the given line is $(1, -1, 3)$. The cross product of these two vectors gives the direction vector of the required line: $(1, 1, 1) \times (1, -1, 3) = (4, -2, -2)$. Thus, the parametric equations of the line are $x = 4t + 3$, $y = -2t + 4$, $z = -2t + 5$, where $(3, 4, 5)$ is the point through which the line passes.

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 $( 3,4,5 )$