Problem Understanding theorm of Cross Product

In summary: You see this kind of thing all the time in physics. You have to be careful and keep track of your units.In summary, the textbook states that the magnitude of the cross product of two vectors is equal to the area of the parallelogram formed by those vectors. While this may seem confusing due to the different units, it is important to note that the numerical values are equal and the dimensions can be consistent if the components of the vectors have dimensions of length. This is a common occurrence in physics and shows the importance of keeping track of units.
  • #1
784
10

Homework Statement


My textbook says:
The length of the cross product a x b is equal to the area of the parallelogram determined by a and b.

How can a length equal and area? They have different units?
 
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  • #2
PsychonautQQ said:

Homework Statement


My textbook says:
The length of the cross product a x b is equal to the area of the parallelogram determined by a and b.

How can a length equal and area? They have different units?
They are numerically equal. That doesn't say anything about units, just that the numbers are the same.
 
  • #3
PsychonautQQ said:

Homework Statement


My textbook says:
The length of the cross product a x b is equal to the area of the parallelogram determined by a and b.

How can a length equal and area? They have different units?

If a and b have dimension length, then axb properly has dimension (length)^2. So does the area of a parallelogram.
 
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  • #4
Dick said:
If a and b have dimension length, then axb properly has dimension (length)^2.
You're going to have to convince me, Dick. a X b is a vector, and its magnitude is a length. I don't see how you can get (length)2 out of a vector.
Dick said:
So does the area of a parallelogram.
 
  • #5
Mark44 said:
You're going to have to convince me, Dick. a X b is a vector, and its magnitude is a length. I don't see how you can get (length)2 out of a vector.

I'll try. The components of axb are products of components of a and b. If the components of a and b have dimension length, that means the components of axb have dimension (length)^2. Not all vectors have dimension length. An example from physics is that angular momentum is defined by l=rxp. r has dimensions of length (m), p has the dimensions of momentum (kg*m/s). l has dimensions of angular momentum (kg*m^2/s).
 
  • #6
Dick, my mind is blown right now.
 
  • #7
Office_Shredder said:
Dick, my mind is blown right now.

I'm kind of surprised this is of mind-blowing proportions. It's just consistently tracking units. In math, you largely treat vectors as dimensionless, so the issue never comes up. If your vectors are dimensionful and you want a length vector perpendicular to a and b, you should use axb/(length units). Which you usually do anyway, just by ignoring the (length)^2 aspect of axb. It should really be satisfying that |axb| correctly has the dimensions of a parallelogram area. Don't get me talking about differential forms and duality. It's why there is a 'cross product' only in three dimensions. The dimension funnyness reflects that.
 
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  • #8
Mark44 said:
You're going to have to convince me, Dick. a X b is a vector, and its magnitude is a length. I don't see how you can get (length)2 out of a vector.
I never use "length" as a synonym for the magnitude of a physical vector. What is the length of a velocity vector? That doesn't quite make sense. Velocity has dimensionality of length per time, not length. The magnitude of that velocity vector? That makes sense, and it even has a name: Speed.
 

1. What is the problem understanding theorem of cross product?

The problem understanding theorem of cross product is a mathematical concept that states that when two vectors are multiplied together using the cross product operation, the resulting vector is perpendicular to both of the original vectors.

2. How does the problem understanding theorem of cross product apply to real-world problems?

The problem understanding theorem of cross product is often used in physics and engineering to calculate the direction and magnitude of a force or torque acting on an object, or to determine the orientation of an object in 3D space.

3. Can the problem understanding theorem of cross product be applied to vectors in any dimension?

Yes, the problem understanding theorem of cross product is applicable to vectors in any number of dimensions. However, it is most commonly used in 3D space.

4. What is the significance of the cross product being perpendicular to the original vectors?

The perpendicular nature of the cross product makes it useful for determining the direction of a vector or the orientation of an object. It also allows for the calculation of torque, which is essential in many engineering and physics problems.

5. Are there any other theorems related to the problem understanding theorem of cross product?

Yes, there are several other theorems related to cross product, such as the right-hand rule and the triple product theorem. These theorems are often used in conjunction with the problem understanding theorem to solve more complex problems.

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