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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.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?
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?
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:If a and b have dimension length, then axb properly has dimension (length)^2.
Dick said:So does the area of a parallelogram.
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.
Office_Shredder said:Dick, my mind is blown right now.
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.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.
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.
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.
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.
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.
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.