# Vectors and how to find the planes to express geometric conditions

• Annabelle1234
In summary: This is the normal of the triangular prism (a triangular prism is a prism with a triangle cross section)
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.

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

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

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.

Annabelle1234 said:
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.

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)

Last edited:
^^ 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!

^^ 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!

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)

## 1. What is a vector?

A vector is a mathematical object that has both magnitude and direction. It is commonly represented by an arrow pointing in the direction of the vector, with the length of the arrow representing the magnitude.

## 2. How are vectors expressed geometrically?

Vectors are typically expressed as a combination of two or three coordinates, depending on the dimension of the space. For example, in two-dimensional space, a vector can be expressed as (x, y), where x and y are the coordinates of the vector's endpoint.

## 3. How do you find the magnitude of a vector?

The magnitude of a vector is calculated using the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In other words, the magnitude of a vector is the square root of the sum of the squared components.

## 4. How do you find the direction of a vector?

The direction of a vector is typically given in terms of angles. In two-dimensional space, the direction can be expressed as an angle measured counterclockwise from the positive x-axis. In three-dimensional space, the direction can be expressed as two angles, one measured from the positive x-axis to the projection of the vector onto the xy-plane, and the other measured from the positive z-axis to the vector.

## 5. How do vectors relate to planes and geometric conditions?

Vectors can be used to represent lines and planes in three-dimensional space. The direction of a vector can determine the orientation of a plane, while the magnitude of a vector can be used to find the distance between the plane and the origin. Additionally, vectors can be used to express geometric conditions, such as parallel or perpendicular lines, and to find the intersection of two planes.

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