Linear Algebra planes in 4d

In summary, the task is to find an equation for a plane in 4-space that is parallel to one given line and contains another given line. In 3D, this can be done by finding a normal vector perpendicular to both lines, but in 4-space, there are an infinite number of such vectors. The attempt at a solution involved taking the dot product of the two given vectors, but this also resulted in an infinite number of possible normal vectors. The student is aware that one point on the plane is given by the line contained in the plane, but is struggling to find another vector to define the plane. However, the student understands that with the additional dimension in 4-space, the space of perpendicular vectors is two-dimensional rather
  • #1
ctrlaltdel121
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Homework Statement



I am given two lines in vector form in 4-space. I need to write an equation for the plane that is parallel to one line, and contains the other line.

Homework Equations


well i know that in 3d I would find a normal vector for the plane that would be perpendicular to both lines, and that would let me define the plane. However i am stumped because in 4-space there are an infinite amount of vectors that are perpendicular to both lines.

The Attempt at a Solution


I took the dot product of the vectors and got an infinite number of possible normals. I know one point on the plane from the line that is contained in the plane, all i need is another vector in the plane to define it but I cannot figure it out.

I didn't want to give the numbers here because i can find it out on my own once I am given some direction.
 
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  • #2
ctrlaltdel121 said:
However i am stumped because in 4-space there are an infinite amount of vectors that are perpendicular to both lines.

The same is true in three dimensions: all the vectors (0, 0, z) for z not equal to 0 are perpendicular to the (x, y) plane.
In this case, the space of vectors which is perpendicular to both lines is just two-dimensional, instead of one. For example, for the (x, y) plane the perpendicular vectors would be (0, 0, z, u) for z.u not equal to 0.
 

1. What is a plane in 4D linear algebra?

A plane in 4D linear algebra is a 3-dimensional subspace that exists within a 4-dimensional space. It is defined by two linearly independent vectors and can be represented by a parametric equation, similar to a plane in 3D.

2. How many dimensions can a plane have in 4D linear algebra?

A plane in 4D linear algebra can have a maximum of 3 dimensions, since it is a subspace within a 4-dimensional space. However, it is possible for the plane to have fewer dimensions, depending on the number of linearly independent vectors used to define it.

3. What is the equation for a plane in 4D linear algebra?

The equation for a plane in 4D linear algebra is similar to the equation for a plane in 3D, but with an additional variable. It can be represented as ax + by + cz + dw = k, where (x, y, z, w) are the coordinates of a point on the plane, (a, b, c) are the direction vectors, and k is a constant.

4. How do you determine if two planes in 4D linear algebra are parallel?

Two planes in 4D linear algebra are parallel if their direction vectors are parallel or if the angle between their normal vectors is 0 or 180 degrees. This can be determined by calculating the dot product of the direction vectors or the cross product of the normal vectors.

5. Can a plane in 4D linear algebra intersect itself?

No, a plane in 4D linear algebra cannot intersect itself. Since it is a subspace, it is defined by unique vectors and cannot contain points that are both on the plane and outside of it. However, it is possible for two different planes in 4D to intersect each other.

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