Cross products in the light of linear transformations | Essence of linear algebra chapter 8 part 2
For anyone who wants to understand the cross product more deeply, this video shows how it relates to a certain linear transformation via duality.
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Hey folks! Where we left off, I was talking about how to compute a three-dimensional cross product between two vectors, v x w. It's this funny thing where you write a matrix, whose second column has the coordinates of v, whose third column has the coordinates of w, but the entries of that first column, weirdly, are the symbols i-hat, j-hat and k-hat where you just pretend like those guys are numbers for the sake of computations. Then with that funky matrix in hand, you compute its determinant. If you just chug along with those computations, ignoring the weirdness, you get some constant times i-hat + some constant times j-hat + some constant times k-hat. How specifically you think about computing that determinant is kind of beside the point. All that really matters here is that you'll end up ...
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Hey folks! Where we left off, I was talking about how to compute a three-dimensional cross product between two vectors, v x w. It's this funny thing where you write a matrix, whose second column has the coordinates of v, whose third column has the coordinates of w, but the entries of that first column, weirdly, are the symbols i-hat, j-hat and k-hat where you just pretend like those guys are numbers for the sake of computations. Then with that funky matrix in hand, you compute its determinant. If you just chug along with those computations, ignoring the weirdness, you get some constant times i-hat + some constant times j-hat + some constant times k-hat. How specifically you think about computing that determinant is kind of beside the point. All that really matters here is that you'll end up ...
