# Another whether a line is parallel to a plane?

• Feb 17th 2013, 01:00 PM
Civy71
Another whether a line is parallel to a plane?
How can you tell whether the line x = x0 + tv in R3 is parallel to the plane x = x0 + t1v1 + t2v2 ?
• Feb 17th 2013, 07:05 PM
chiro
Re: Another whether a line is parallel to a plane?
Hey Civy71.

Hint: If a line lies on a plane then the line will be perpendicular to the plane normal.
• Feb 17th 2013, 07:12 PM
Plato
Re: Another whether a line is parallel to a plane?
Quote:

Originally Posted by Civy71
How can you tell whether the line $\displaystyle x = x_0 + tv$ in R3 is parallel to the plane $\displaystyle x = x_0 + t_1v_1 + t_2v_2$ ?

Surely both line and plane have $\displaystyle x_0$ in common.

Question is $\displaystyle v\cdot(t_1\times t_2)=0~?$
• Feb 18th 2013, 06:28 AM
Civy71
Re: Another whether a line is parallel to a plane?
yes, it equals 0
• Feb 18th 2013, 06:48 AM
Plato
Re: Another whether a line is parallel to a plane?
Quote:

Originally Posted by Civy71
yes, it equals 0

Well then, a line with direction vector $\displaystyle D$ is parallel to a plane with normal $\displaystyle N$ if and only if $\displaystyle D\cdot N=0$.
• Feb 18th 2013, 09:41 AM
johng
Re: Another whether a line is parallel to a plane?
Here's another way to view the solution, purely algebraic.

Attachment 27114
• Feb 18th 2013, 10:28 AM
Hartlw
Re: Another whether a line is parallel to a plane?
Quote:

Originally Posted by Civy71
How can you tell whether the line x = x0 + tv in R3 is parallel to the plane x = x0 + t1v1 + t2v2 ?

v.v1Xv2 = 0. Same as det |v,v1,v2 | = 0 , ie v, v1 and v2 are linearly dependent.

EDIT: v is a vector in the direction of the line. v1 and v2 are vectors in the plane.