Are two vectors that are orthogonal to a third parallel?

In summary, the conversation is discussing whether or not it is true in three dimensions that any two vectors that are perpendicular to a third vector are also parallel to each other. The use of dot product and cross product are suggested as methods to find a counterexample to prove the statement false. The concept of the right hand rule is also mentioned as a way to visualize the problem. Ultimately, it is advised to disregard the post and focus on finding a counterexample using the given information.
  • #1
Vitani11
275
3

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?
 
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  • #2
You're probably overthinking the problem. You should be able to easily come up with a counterexample which shows the statement is false.
 
  • #3
Not in general, consider the dot products of two non-parallel vectors with the 0 vector.
 
  • #4
If you want something visual, you might ponder the right hand rule...
 
  • #5
Vitani11 said:
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.
 
Last edited:
  • #6
Vitani11 said:
Is it true in three dimensions that any two vectors perpendicular to a third one are parallel to each other?
XYZ axes?
 

FAQ: Are two vectors that are orthogonal to a third parallel?

1. Are two vectors that are orthogonal to a third parallel?

No, two vectors that are orthogonal to a third are not necessarily parallel. Orthogonal vectors are perpendicular to each other, meaning they form a 90-degree angle. Parallel vectors, on the other hand, have the same direction and may or may not have the same magnitude.

2. What does it mean for two vectors to be orthogonal?

Orthogonal vectors are two vectors that are perpendicular to each other, meaning they form a 90-degree angle. This also means that their dot product is equal to zero.

3. Can two vectors be orthogonal but not parallel?

Yes, two vectors can be orthogonal without being parallel. Orthogonal vectors are perpendicular to each other, meaning they form a 90-degree angle. Parallel vectors, on the other hand, have the same direction and may or may not have the same magnitude.

4. How can I determine if two vectors are orthogonal?

To determine if two vectors are orthogonal, you can use the dot product formula: A · B = |A||B|cosθ. If the dot product is equal to zero, the vectors are orthogonal. You can also visually inspect the two vectors and check if they form a 90-degree angle.

5. What is the geometric interpretation of orthogonal vectors?

The geometric interpretation of orthogonal vectors is that they are perpendicular to each other, forming a 90-degree angle. This can be visualized as the two vectors intersecting at a right angle on a coordinate plane.

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