Question

Find a vector equation and parametric equations for the line. (Use the parameter t.) The line through the point (1, 0, 9) and perpendicular to the plane x 2y z

Answer 1

the parameter equation of given line is

x= 1+t

y=0+ 2t

z= 9+t

( note: in plane no signs given so we assume + so x+2y+z)

Explanation:

A vector perpendicular to the plane a x + b y + c z + d = 0 is given by ⟨ a , b , c ⟩

So a vector perpendicular to the plane x +2 y + z − = 0 is ⟨ 1 , 2 , 1 ⟩

The parametric equation of a line through
`(x_(0) ,y_(0), z_(0) )` and parallel to the vector ⟨ a , b , c ⟩ is

`x=x_(0) +ta\\y=y_(0) +tb\\z=z_(0) +tc`

so the parametric equation of our line is

`x=1+t\\y=2t\\z=9+t`

The vector form of the line is

`r=<1,0,9>+t<1,2,1>`