Cross product and matrix multiplication

The correct statement is that if M is an orthogonal matrix with determinant 1, then (Ma) x (Mb) = Mc.In summary, when dealing with a cross-product of vectors in a new coordinate system, it is important to ensure that the orthogonal matrix used has a determinant of 1. This preserves the norms of the vectors and maintains the right-hand rule, making the statement (Ma) x (Mb) = Mc true. If the determinant is -1, then the orientation is changed and the statement becomes false.
  • #1
haael
539
35
Suppose that we have a cross-product of vectors.

[tex]a × b = c[/tex]

Now suppose that we have an orthogonal matrix M. Is it true that

[tex](M a) × (M b) = M c[/tex]

?

My intuition is that here we are moving to another coordinate system and performing a cross product in this new system. I can't find an answer in google, so I'm posting here.
 
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  • #2
Give it determinant 1, otherwise it will reverse orientation and screw up your right-hand rule. Then, it works. The transformation will preserve the norms of the vectors, and it will also map orthogonal vectors to orthogonal vectors. So, far that leaves only two possibilities when you take the cross product. Which is why you make it have determinant one. Then, the right hand rule is taken care of.
 
  • #3
It looks like proof goes through if you check (M_1a_1+...)x(M_1b_1+...), distribute (like FOIL) and use orthonormality of columns of M. I didnt check if the determinant = 1 condition mentioned above is necessary.
 
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  • #4
algebrat said:
I didnt check if the determinant = 1 condition mentioned above is necessary.

It should be necessary. If the determinant is -1, then it is false since orientation is changed.
 
  • #5


I can confirm that your intuition is correct. When we perform a cross product, we are finding the vector that is perpendicular to both vectors a and b. This vector, c, will have a magnitude equal to the area of the parallelogram formed by vectors a and b, and its direction will follow the right-hand rule.

When we apply an orthogonal matrix, we are essentially rotating our coordinate system. This means that the vectors a and b will also be rotated, and their cross product will result in a new vector, c', that is perpendicular to both the rotated vectors (Ma) and (Mb). However, the magnitude and direction of c' may not be the same as c, as they are now in a different coordinate system.

Therefore, it is not necessarily true that (Ma) × (Mb) = Mc. The cross product operation is dependent on the coordinate system, and applying an orthogonal matrix will change the coordinate system and therefore change the resulting vector. However, it is possible for (Ma) × (Mb) to be equal to Mc in certain cases, such as when the orthogonal matrix is the identity matrix and does not change the coordinate system.

In conclusion, the cross product and matrix multiplication are two separate operations and their results may not always be equivalent. It is important to consider the coordinate system and any transformations applied when performing these operations.
 

1. What is the difference between cross product and matrix multiplication?

Cross product is a vector operation that results in a new vector that is perpendicular to the original two vectors. Matrix multiplication is a mathematical operation that combines two matrices to produce a new matrix.

2. Can cross product and matrix multiplication be performed on any type of matrix or vector?

No, cross product can only be performed on two three-dimensional vectors, while matrix multiplication can be performed on any compatible matrices. For example, a 3x2 matrix can be multiplied with a 2x4 matrix, resulting in a 3x4 matrix.

3. How is the cross product calculated?

The cross product is calculated by taking the determinant of a 3x3 matrix with the first row being the unit vectors (i, j, k), the second row being the components of the first vector, and the third row being the components of the second vector.

4. What are some applications of cross product and matrix multiplication in science?

Cross product is commonly used in physics and engineering to calculate torque, angular momentum, and magnetic fields. Matrix multiplication is used in a variety of fields, such as computer science, economics, and statistics, for data analysis and modeling.

5. Is it possible to perform cross product and matrix multiplication simultaneously?

No, it is not possible to perform cross product and matrix multiplication at the same time as they are two distinct operations with different rules and calculations. However, the result of one operation can be used as an input for the other. For example, the cross product of two vectors can be used as a scaling factor in matrix multiplication.

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