Do Orthogonal Vectors in 3-D Always Lie on a Straight Line?

[war485]

Homework Statement

The problem is more of a concept/visual problem, like to find out if it's true or false and why:

all vectors orthogonal to a non zero vector in R^3 are contained in a straight line.

Homework Equations

No equations, just very bad at visualizing in 3-D

The Attempt at a Solution

I know that for a vector to be orthogonal, it needs to be at 90 degrees with another (non zero vector). So, when I visualize it in my head, I see a straight line ( a vector) going out in the middle of a 3-D cube from the middle. Then, if I take a 90 degrees anywhere along the line, I see a "normal" 90 degree line, but I can rotate that normal line all around that point, so I think I should always get a plane and not a line, but I'm not sure if all vectors are like that because I don't know if a vector in R^3 can be a line like that. Or, if I had a plane and then a "normal" plane going at it at 90 degrees, then all of those "normals" also form a plane. So, I think that it's true that ALL vectors orthogonal to a non zero vector in R^3 are contained in a straight line.

I'm in my first year at college in Matrix Theory/Linear algebra.

 

[HallsofIvy]

Homework Statement

The problem is more of a concept/visual problem, like to find out if it's true or false and why:

all vectors orthogonal to a non zero vector in R^3 are contained in a straight line.

Homework Equations

No equations, just very bad at visualizing in 3-D

The Attempt at a Solution

I know that for a vector to be orthogonal, it needs to be at 90 degrees with another (non zero vector). So, when I visualize it in my head, I see a straight line ( a vector) going out in the middle of a 3-D cube from the middle. Then, if I take a 90 degrees anywhere along the line, I see a "normal" 90 degree line, but I can rotate that normal line all around that point, so I think I should always get a plane and not a line, but I'm not sure if all vectors are like that because I don't know if a vector in R^3 can be a line like that.

Yes, a vector can be visualized as a (directed) line segment.

Or, if I had a plane and then a "normal" plane going at it at 90 degrees, then all of those "normals" also form a plane.

?Why would you be thinking about a plane? You were asked about normals to a line.

So, I think that it's true that ALL vectors orthogonal to a non zero vector in R^3 are contained in a straight line.

I'm in my first year at college in Matrix Theory/Linear algebra.

You have given two reasons why the set of all vectors perpendicular to a given line form a plane, not a line but then say you are not sure and immediately conclude that they must form a line! Are you saying that if you are not sure you are right, then you must be wrong? How about a little self confidence?

 

[war485]

So, I think that it's true that ALL vectors orthogonal to a non zero vector in R^3 are contained in a straight line.

I just realized a small typo between here and what I wrote on paper, forgot to say that "it's not true", and must have left out the "not" part of it on-line (otherwise my reasoning doesn't make sense). I think I see that the original statement is false for vectors of straight lines because it forms a plane and not a straight line. But then I was also thinking about planes because I was trying to see if it would be true for other vectors in R^3, so would it be ok to generalize like that to all vectors?

 

[Defennder]

Well first of all what do you understand by "vector"? You seem to be under the impression that a vector is necessarily attached to geometrical objects such as lines by your statement "vectors of straight lines".

 

[CompuChip]

You can think of a vector as just a point in space. A vector can be represented by its coordinates (x, y, z). It can also be represented by an arrow pointing from the origin (the point which has by definition coordinates (0, 0, 0)) to that point.

A vector spans a line, consisting of all the points (x', y', z') whose coordinates satisfy (x', y', z') = c (x, y, z) = (cx, cy, cz) for a constant c.

All the vectors orthogonal to a vector in n-dimensional space, form an (n-1)-dimensional space. For n = 3: the vectors orthogonal to some fixed vector form a plane. In fact, this is sometimes used in physics. For example, instead of writing down a plane of rotation we can write down any vector orthogonal to that plane, so instead of two vectors spanning the plane we just have to give one vector orthogonal to the plane to uniquely define which plane we mean (and in the case of a rotation, since the magnitude is irrelevant to define the plane, we can use the magnitude to indicate the angle of rotation).

Anyway, as Defender said, you seem to be a bit confused about what vectors are. You can think of them just as points or arrows and you should also discern between vectors forming some space (for example, a line) and just spanning it. For example, the vector (1, 0, 0) spans the x-axis, but it does not form the x-axis; for that you need the set of (x, 0, 0) for all x in R.

 

[war485]

I never knew vectors can be considered as points, I always associated them with arrows pointing from the origin or from the head of another arrow, like in physics. But since vectors can be a point by its coordinates or an arrow from the origin, then it wouldn't make sense to say that a vector in 3-d (as a point) has an orthogonal vector in a plane, does it?

thanks for the help on clearing it up.

 

[HallsofIvy]

Has an orthogonal vector in a plane? Any vector has an infinite number of vectors perpendicular to it. If the vector is in a space of dimension n, then the vectors perpendicular to it span a space of dimension n-1. In particular, if n= 3, n-1= 2.