# How to determine whether a line lies in / is parallel to / intersects a plane?

1. Jul 21, 2012

### NeroGoh

Referred to the orgin O,the positon vectors of the point A and B are (i+j+3k) and ( 4i-2j+5k) respectively.The equation of the plane π is x+2y-z=10.
Show that the point A and B do not lie on the plane.

Last edited: Jul 21, 2012
2. Jul 21, 2012

### Ibix

3. Jul 21, 2012

### LCKurtz

What you are saying to show here does not match the problem statement in your title.

4. Jul 21, 2012

### HallsofIvy

Staff Emeritus
To determine if line l lies in plane P, replace x, y, and z in the equation of the plane with their expressions in terms of some parameter. That will give a single linear equation for that parameter. If the equation is satisfied by a single value of the parameter, the line intersects the plane and that value of the parameter gives the point of intersection. If NO value of the parameter satisfies the equation, the line is parallel to the plane. If the equation is satisfied for every member of the parameter, the line lies in the plane.

As for the problem you actually posted, if the position vector of a point is Ai+ Bj+ Ck, then the point is (x, y, z)= (A, B, C). Put those into the equation. If it is satisfied then the point lies in the plane. If not, the point is not in the plane.

Last edited: Jul 22, 2012
5. Jul 22, 2012

### NeroGoh

my solution dunno whether correct or wrong help me if wrong... ty ^^

i take the plane,
x+2y-z=10 => x+2y-z-10=0.

and then i take the point A and B put in the plane equation...
A(1,1,3): x+2y-z-10=0
1+2(1)-3-10=0
-10≠0
*point A do not lie on the plane

B(4,-2,5): x+2y-z-10=0
4+2(-2)-5-10=0
-15≠0
*pont B do not lie on the plane

izzit correct???? /.\

6. Jul 22, 2012

### HallsofIvy

Staff Emeritus
Yes, that is correct.

7. Jul 22, 2012

### LCKurtz

But, of course, you haven't determined whether the line is parallel to the plane or intersects the plane which, according to the title, is what you were asked to determine. Or is that not what you were trying to do?