Are two vectors that are orthogonal to a third parallel?

[Vitani11]

Homework Statement

Is it true in three dimensions that any two vectors perpendicular to a third one are parallel to each other?

Homework Equations

Dot product.

The Attempt at a Solution

I've come up with two vectors that were orthogonal to a third and found the angle between them using the definition of the dot product and the angle was not 180 degrees. Therefore I don't think that it's true. I'm really only here to check that I did my math right. Is it actually not true, or do I need to recalculate?

 

[vela]

You're probably overthinking the problem. You should be able to easily come up with a counterexample which shows the statement is false.

 

[VrhoZna]

Not in general, consider the dot products of two non-parallel vectors with the 0 vector.

 

[StoneTemplePython]

If you want something visual, you might ponder the right hand rule...

 

[Eclair_de_XII]

Dot product

In my opinion, you should try computing the cross product of two parallel vectors, since the cross product produces a vector normal to both those vectors. Can you do it? If you can answer that question, then you can answer the original question, I think. Given you know how to cross-multiply vectors. But since you're given only the definition of the dot product, you can kindly disregard this post.

 

[haruspex]

Is it true in three dimensions that any two vectors perpendicular to a third one are parallel to each other?

XYZ axes?