Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The cross product $a \times b$ can be represented by the determinant

$$\mathbf{a}\times\mathbf{b}= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix}.$$

Does the matrix whose determinant is this have any significance?

share|improve this question
2  
It seems doubtful, given that it's only a matrix in a certain formal sense, with different elements in the first row than in the second and third rows. One might expect a matrix with further significance to correspond to some linear transformation, but for that it would have to be a "proper" matrix. –  joriki Jun 5 '12 at 11:45
    
I agree with joriki that it is mainly just a helpful mnemonic device to remember how to compute the product, and not a real matrix. –  rschwieb Jun 5 '12 at 11:57
add comment

1 Answer 1

up vote 2 down vote accepted

You can let the matrix act by ordinary matrix multiplication on ordinary vectors in three-dimensional space.

This will transform a vector in a triple containing the original vector and the lengths of the two projections on $a$ and $b$.

While I feel that this counts as "any significance", it isn't very satisfactory, because the matrix, as you presented it, does not allow for matrix multiplication.

I think it is a much more useful point of view, to first view $i$,$j$,$k$ as three scalar variables (better denoted by $x$, $y$, $z$), then take the determinant of your matrix and then regard the cross product as the gradient vector this determinant.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.