Cross Product of Parallel Vectors is the zero vector (why?)

[Brandon Hawi]

Hello, PF!

I had a quick question that I hoped maybe some of you could help me answer. The question is simple: Why is the cross product of two parallel vectors equal to the zero vector? I can see this easily mathematically through completing the cross product formula with two parallel vectors, but I wanted to know why this existed. How does this fit into the definition of a cross product? To my knowledge, in simple terms, the vector you get from a cross product operation results in a vector perpendicular to both the vectors. Anyway, if anyone could help explain this, feel free to in the thread.

Thanks!

 

[jedishrfu]

Look at the original definition for cross product AxB = |A| * |B| * sin(AB angle)

What can you determine if A and B are not the zero vector but AxB = 0?

EDIT: So you understood that A could be parallel to B and hence AxB = 0

In that case, there would be a whole plane of vectors that are perpendicular to A and B.

 

[Brandon Hawi]

Look at the original definition for cross product AxB = |A| * |B| * sin(AB angle)

What can you determine if A and B are not the zero vector but AxB = 0?

Wouldn't that have to mean that the angle between is either 0 or pi or any multiple of pi?

 

[jedishrfu]

If its zero or any multiple of PI then the vectors are parallel or anti-parallel (ie parallel but pointing in opposite directions)

 

[Brandon Hawi]

If its zero or any multiple of PI then the vectors are parallel or anti-parallel (ie parallel but pointing in opposite directions)

I understand this, and can see this in all the formulas and such that are related to the cross product. But I can't wrap my head around it conceptually. Shouldn't there be a ton of different vectors that are perpendicular to both of the parallel vectors?

 

[jedishrfu]

Yes, that's what I said there's a whole plane of vectors that are perpendicular to A and to B (see post #2 I added a line while you had posted yours).

So basically there's no unique vector and so its the zero vector.

 

[Brandon Hawi]

Yes, that's what I said there's a whole plane of vectors that are perpendicular to A and to B (see post #2 I added a line while you had posted yours).

So basically there's no unique vector and so its the zero vector.

OH okay! So is the best way to represent the whole plane of vectors to use the zero vector?

 

[jedishrfu]

Try not to make broad statements like that though.

If the two vectors A and B are parallel or anti-parallel or A or B is the zero vector then you get by definition the zero vector.

I say this because a plane is often defined using a vector that's normal to it.

Here's some more info on cross-products:

https://en.wikipedia.org/wiki/Cross_product

 

[Brandon Hawi]

Try not to make broad statements like that though. If the two vectors A and B are parallel or anti-parallel or A or B is the zero vector then you get by definition the zero vector.

I say this because a plane is often defined using a vector normal to it.

Here's some more info on cross-products:

https://en.wikipedia.org/wiki/Cross_product

Alright thanks so much!

 

[Merlin3189]

How about thinking of a specific case?

Torque is the cross product of the force vector and the distance vector from the axis. If the force is applied parallel to the axis, there is zero torque.

The force on a current carrying conductor in a magnetic field is the cp of the current and the magnetic field vectors. If they are parallel, there is no force.If on the other hand you are thinking about vectors in abstract (mathematically?) then what other answer can there be than the definition, AxB = |A| * |B| * sin(AB angle)?
What meaning would YOU ascribe to the cross product of parallel vectors?

 

[Mark44]

Look at the original definition for cross product AxB = |A| * |B| * sin(AB angle)

If on the other hand you are thinking about vectors in abstract (mathematically?) then what other answer can there be than the definition, AxB = |A| * |B| * sin(AB angle)?

Technically speaking, |A| * |B| * sin(AB angle) represents the magnitude of the cross product -- |A x B| -- not the actual cross product.

 

[robphy]

To my knowledge, in simple terms, the vector you get from a cross product operation results in a vector perpendicular to both the vectors.

In my opinion, in a cross-product, more emphasis needs to be placed on
the oriented-parallelogram formed from the given pair of vectors [with their tails together, or with the second placed at the tip of the first].
The magnitude of the cross-product is the magnitude of the parallelogram's area.
The direction of the cross-product is the perpendicular [a.k.a. "normal"] to that parallelogram, using the right-hand rule.

So, with a pair of parallel vectors, its parallelogram has zero area.