How Does a Line Intersect with Planes in Vector Geometry?

[otterandseal1]

Homework Statement

I'm struggling to get part b. I got part a to be (1,4,-1). Just got no clue for b. Pls help

Relevant Equations

Dont know what to put here

 

[Mark44]

Homework Statement:: I'm struggling to get part b. I got part a to be (1,4,-1). Just got no clue for b. Pls help
Relevant Equations:: Dont know what to put here

View attachment 259080

Can you find a normal to $\pi_1$? You should be able to do this by inspection from its equation. Also, if two vectors are perpendicular, their dot product is 0.

 

[otterandseal1]

Can you find a normal to $\pi_1$? You should be able to do this by inspection from its equation. Also, if two vectors are perpendicular, their dot product is 0.

Is the normal (2,-1,-3)?

 

[ehild]

Is the normal (2,-1,-3)?

Yes.
You can find the normal of the other plane if you know two vectors the plane contains.

 

[otterandseal1]

Yes.
You can find the normal of the other plane if you know two vectors the plane contains.

is that using r.n = a.n ?

 

[Mark44]

is that using r.n = a.n ?

If $\vec a$ and $\vec b$ are vectors that lie in the plane you're trying to find the equation of, then $\vec a \times \vec b$ gives you a normal to that plane.

 

[otterandseal1]

If $\vec a$ and $\vec b$ are vectors that lie in the plane you're trying to find the equation of, then $\vec a \times \vec b$ gives you a normal to that plane.

Could you possibly show me the working for the question please?

 

[Mark44]

Could you possibly show me the working for the question please?

No, can't do that, per the forum rules. See https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/, under Homework Guidelines.

Can you find two vectors in the plane you're trying to find?
If so, can you find their cross product? That will give you a normal to that plane. Once you have the normal, all you need is a single point in the plane, and you can find its equation.

 

[otterandseal1]

No, can't do that, per the forum rules. See https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/, under Homework Guidelines.

Can you find two vectors in the plane you're trying to find?
If so, can you find their cross product? That will give you a normal to that plane. Once you have the normal, all you need is a single point in the plane, and you can find its equation.

Could I use the vector from part a? and the normal of the first plane?

 

[ehild]

Could I use the vector from part a? and the normal of the first plane?

What vector do you mean? And, yes. as the two planes are perpendicular the second plane contains the normal of the first plane.

 

[otterandseal1]

What vector do you mean? And, yes. as the two planes are perpendicular the second plane contains the normal of the first plane.

Where the line meets the first plane

 

[ehild]

Where the line meets the first plane

No, it is a point , common point of the first plane an the line I. P (1,4,-1) is the position vector a of this point, it does not lie in the plane as vector b. But the line I lies in the second plane and so is its direction vector. What is it?