# Thread: Vector Problem - Plane and normal

1. ## Vector Problem - Plane and normal

Find a vector that is normal to the plane containing the lines L1 and L2, whose equations are

L1 = r = i + k + λ(2i + j - 2k)
L2 = r = 3i + 2j + 2k + μ(j + 3k)

2. Can you show that the point $\displaystyle (3,1,-1)$ is on both lines?

The the normal you need is $\displaystyle <2,1,-2>\times <0,1,3>$.

3. Oh so i find the cross product of <2,1,2> and <0,1,3>?

Im not sure how to do this bit --- Can you show that the point is on both lines?

4. Originally Posted by Jampop
Oh so i find the cross product of <2,1,-2> and <0,1,3>?

Im not sure how to do this bit --- Can you show that the point is on both lines?
Show the lines intersect at that point.

5. Oh ok, but why is significant?

6. Originally Posted by Jampop
Oh ok, but why is significant?
If the lines are do not intersect and are not parallel there is no plane that contains both lines. So the two must intersect.
I will not do that algebra for you.

7. Ive shown that they intersect at <3,1,-1> , is this the vector that is normal to the plane?

8. Originally Posted by Jampop
Ive shown that they intersect at <3,1,-1> , is this the vector that is normal to the plane?
No indeed. $\displaystyle (3,1,-1)$ is the point we use to write the equation of the plane with normal $\displaystyle <2,1,-2>\times <0,1,3>$.

9. But how do i Find a vector that is normal to the plane ??? Is it just <2,1,-2> cross <0,1,3>