Planes and lines...

Ok im confused ,

If i want a plane parrarlel to another plane and containing a line i know i just use the normal vector of the plane and dot product it with a point on the line to get r.n=d

However how does simply dot producting the normal vector with a point on the line ensure the line is now contained in the plane , im confused...

Reply 1

ttoby

15

It doesn't. You'll ensure that at least one point from the line is contained in the plane but there is the potential for the line to just pass straight through the plane, in which case it would be impossible for the plane to be adjusted so that it contains the whole line and is still parallel to the plane given in the question.

In short, this method only works for certain combinations of lines and planes, and if the combination given is not nice then there would be no solution to the question.

Edit: Here's an example of when it wouldn't work: Find a plane that is parallel to the x-z plane but also contains the y axis.

(edited 12 years ago)

Reply 2

falcon pluse

OP

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Reply 3

falcon pluse

OP

Original post by ttoby

It doesn't. You'll ensure that at least one point from the line is contained in the plane but there is the potential for the line to just pass straight through the plane, in which case it would be impossible for the plane to be adjusted so that it contains the whole line and is still parallel to the plane given in the question.

In short, this method only works for certain combinations of lines and planes, and if the combination given is not nice then there would be no solution to the question.

Edit: Here's an example of when it wouldn't work: Find a plane that is parallel to the x-z plane but also contains the y axis.

OK so to clear this up , it only works for certain combinations, just curious as in my exam questions it just seemed so odd that it worked.

Could you also help explain this , i did a question of finding a plane (p2)perpendicular to a plane (p1) and containing a line which passed through p2. So by doing the cross product of the normal vector of p1 and direction vector of the line i get a normal vecotor but my dotting it with the line, there is a potential the line could just pass throught the plane?, it only worked with the combo i was given?

Reply 4

ttoby

15

Original post by falcon pluse

OK so to clear this up , it only works for certain combinations, just curious as in my exam questions it just seemed so odd that it worked.

Could you also help explain this , i did a question of finding a plane (p2)perpendicular to a plane (p1) and containing a line which passed through p2. So by doing the cross product of the normal vector of p1 and direction vector of the line i get a normal vecotor but my dotting it with the line, there is a potential the line could just pass throught the plane?, it only worked with the combo i was given?

The thing you need to consider is whether the line you want in your plane is perpendicular to the plane's normal.

If they are perpendicular to each other then this means that the line must be parallel to the plane, and so you can get it to be in the plane if you have the plane in the right position. If the line and normal are not perpendicular then the line would just pass through the plane.

You can check which case it is by taking the dot product of the line and the normal.

In your exam questions, it's likely that they just chose a combination that worked.

For your example about the planes p1 and p2, when you took the cross product, the vector you got out would have been perpendicular to both p1's normal and the line. So this means that the normal to p2 would be perpendicular to the line, and hence the line cannot just pass through the plane.

Planes and lines...

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