# Thread: Another whether a line is parallel to a plane?

1. ## 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 ?

2. ## 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.

3. ## Re: Another whether a line is parallel to a plane?

Originally Posted by Civy71
How can you tell whether the line $x = x_0 + tv$ in R3 is parallel to the plane $x = x_0 + t_1v_1 + t_2v_2$ ?
Surely both line and plane have $x_0$ in common.

Question is $v\cdot(t_1\times t_2)=0~?$

4. ## Re: Another whether a line is parallel to a plane?

yes, it equals 0

5. ## Re: Another whether a line is parallel to a plane?

Originally Posted by Civy71
yes, it equals 0
Well then, a line with direction vector $D$ is parallel to a plane with normal $N$ if and only if $D\cdot N=0$.

6. ## Re: Another whether a line is parallel to a plane?

Here's another way to view the solution, purely algebraic.

7. ## Re: Another whether a line is parallel to a plane?

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.