Vectors and how to find the planes to express geometric conditions

[Annabelle1234]

Homework Statement

The normal vector of each of the following planes is determined from the coefficients of the x-, y-, z- terms.
pi1: a1x+b1y + c1z + d1=0
pi2: a2x+b2y+c2z+d2=0
pi3: a3x+b3y+c3z+d3=0

Define the extended vector for each plan as follows:

e1= [a1, b1, c1, d1]
e2= [a2, b2, c2, d2]
e3=[a3, b3, c3, d3]
Use the extended (where necessary) and normal vectors of these planes to express the following geometric conditions:

a) 3 parallel, distinct planes
b) 2 parallel distinct planes intersected by another plane to form 2parallel lines
c) 3 distinct planes intersecting in a line
d) 3 distinct planes forming a triangular prism, that is, no common points of intersection but intersecting in pairs to form 3 parallel lines
e)3 distinct planes intersecting in a unique point

The Attempt at a Solution

I am not really sure how to go about doing this. I tried simply trying to plug in numbers for the variable but i was unsuccessful. The examples in the textbook also doesn't help much. I would really appreciate it if someone could guide me onto how I could start this.

This is the first time I've posted a question here; I really hope you guys can help me solve this.

Thanks in advance! =)

 

[lanedance]

well for a), they will have the same normal vector(a,b,c) as they are parallel but a different d value as they are distinct

 

[lanedance]

the trick to these is visualizing what they're asking, try drawing pictures and see if they line up with the questions

 

[Annabelle1234]

So I shouldn't try to substitute numbers?

I had been trying to do that all along. Okay... I'll try drawing it down instead. Thanks so much... I hope I'd be able to solve this problem now.

 

[Quinzio]

So I shouldn't try to substitute numbers?

I had been trying to do that all along. Okay... I'll try drawing it down instead. Thanks so much... I hope I'd be able to solve this problem now.

What do you exactly mean by "substituting numbers" ?
They're not asking you to find a particular solution, but to find some equations, some relations, that link the coefficients together to satisfy the problem requirements.

It looks like, pardon if I'm wrong, you are no so familiar with 3d geometry, that is plane equations, normal vectors, dot product, cross product, ecc...
Unless you are familiar with this concept, you cannot answer the problem.

 

[lanedance]

you want to find relations between the ai, bi, ci, di for each of the "extended" plane vectors

in problem a) it is
a_1 = a_2 = a_3
b_1 = b_2 = b_3
c_1 = c_2 = c_3
d_1 \neq d_2, \ d_2 \neq d_3, \ d_3 \neq d_1

which expresses the plane normals are equivalent (though only required up to a scalar multiplicative constant), and intercepts are different

to write the multiplicative constants correctly (as the plane equations are equivalent when multiplied by a constant ) it should be for some scalars \lambda, \ \mu
a_1 = \lambda a_2 = \mu a_3
b_1 = \lambda b_2 = \mu b_3
c_1 = \lambda c_2 = \mu c_3
d_1 \neq \lambda d_2, \ \lambda d_2 \neq \mu d_3, \ \mu d_3 \neq d_1

otherwise you could assume the normal vectors are normalised (length=1)

 

[Annabelle1234]

^^ I was trying to substitute numbers because working in variables was confusing. I do know what cross product, dot product etc. means, I am just not sure how to apply them in solving any of these problems.

I think I know how to do 'c' and 'e' but I am not sure how to go about doing 'b' and 'd' because it asks to satisfy a lot criteria as opposed to simply finding parallel/intersecting lines.

@Lanedance - Thank you for the formula. It really helped!

 

[Annabelle1234]

^^ I was trying to substitute numbers because working in variables was confusing. I do know what cross product, dot product etc. means, I am just not sure how to apply them in solving any of these problems.

I think I know how to do 'c' and 'e' but I am not sure how to go about doing 'b' and 'd' because it asks to satisfy a lot criteria as opposed to simply finding parallel/intersecting lines.

@Lanedance - Thank you for the formula. It really helped!

 

[lanedance]

b) you have 2 planes parallel (same normal direction) but distinct (different intercept). The third plane is not parallel (different normal - cross product with others is non-zero)

d) you really need to visualise this one to understand it. First imagine 3 parallel but distinct lines, now put 3 non-parallel planes (different normals). One containing each line. now as they each contain the same line direction, the normal of each plane will be perpindicular to the line direction (dot product is zero)