Cross-products and the right-hand rule

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In summary, torque represents a rotation about an axis and the torque vector points in the direction of the axis. A cross product is the measurement of the amount of torque about an axis caused by two vectors and can also be seen as the measurement of perpendicularity between the two vectors. The determinant of a 3x3 matrix represents the volume formed by 3 vectors, with the area of a parallelogram being the volume when one vector is the unit vector. The choice of direction for the resultant of a cross product is not arbitrary and depends on the chosen coordinate system. In a right-handed coordinate system, the right-hand rule is used while in a left-handed coordinate system, the left-hand rule is used to find the resultant direction.
  • #1
Kavorka
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I understand how to do problems relating to this, I just don't really understand the concept behind it.

Why is the torque vector in rotational motion directed out of the plane of rotation? I can't wrap my head around why it would be directed that way in a real-world set of mind.

What does a cross-product actually represent? I understand that its magnitude is the area of a parallelogram formed by the two crossed vectors. I'm not sure why the determinant of a 3x3 matrix actually means something however. I know what it means in terms of mathematics, but not in terms of real-world forces and physics that I can conceptualize.
 
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  • #2
A torque represents a rotation about an axis. So the vector is just pointing in the direction of the axis.

A cross product is the amount of torque about an axis that two vectors cause where the axis can be in any direction. It can also be considered a measurement of the amount of perpendicularity between two vectors while the dot product is the measurement of the amount of parallelism.

A determinant of a 3x3 matrix generally can be thought of as the volume that 3 vectors form. So if one of the vectors is the unit vector then the volume becomes the area of a parallelogram.
 
  • #3
Hm that makes more sense. Is the choice of the vector pointing in or out when the force is applied clockwise or counterclockwise just arbitrary?
 
  • #4
Completely.
 
  • #5
Kavorka said:
Hm that makes more sense. Is the choice of the vector pointing in or out when the force is applied clockwise or counterclockwise just arbitrary?

It depends upon your choice of coordinate system. Cartesian coordinate systems can be right-handed (the usual case) or left-handed. In a right-handed coordinate system you use the right-hand rule to find the resultant direction of a cross product, and ##\hat{i} \times \hat{j} = \hat{k}##. In a left-handed coordinate system ##\hat{i} \times \hat{j} = -\hat{k}## and you'd use the left-hand rule to find the direction.

For a given coordinate system the direction of the resultant of a cross product of two vectors is not arbitrary. If it were arbitrary, conservation laws like conservation of angular momentum would not hold because the addition of angular momentum vectors would then be arbitrary, too, given arbitrary choices for their directions.
 
  • #6
So is the OP right-handed or left-handed?
 
  • #7
right-handed of course!
 

1. What is a cross product?

A cross product is a mathematical operation that takes two vectors as inputs and produces a third vector that is perpendicular to both of the input vectors.

2. How do you calculate a cross product?

To calculate a cross product, you first need to determine the direction of the resulting vector using the right-hand rule. Then, you can use the formula:
A x B = (Ay*Bz - Az*By)i + (Az*Bx - Ax*Bz)j + (Ax*By - Ay*Bx)k
where A and B are the two input vectors.

3. What is the right-hand rule?

The right-hand rule is a method used to determine the direction of the resulting vector in a cross product. It states that if you point your right thumb in the direction of the first vector and your index finger in the direction of the second vector, then your middle finger will point in the direction of the resulting vector.

4. When is a cross product used?

A cross product is often used in physics and engineering to calculate the torque or angular momentum of an object. It is also used in computer graphics to determine the direction of a normal vector to a surface.

5. Can a cross product be negative?

Yes, a cross product can be negative. The negative sign indicates that the resulting vector is pointing in the opposite direction of what the right-hand rule predicts. This is often referred to as a "left-handed" cross product.

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