this is a much intuitive explanation than the diagonal rule
Heisenberg
The geometic explanation really helps with the understanding of Triple Product
Osoii
I think a little bit more about why this formula is the volume: from the geometric definition, (u cross v).w = |u cross v||w|cos, where |u cross v| is the area of the base, and |w|cos is the height of the parallelepiped since u cross v is orthogonal to the base.
degrees_K
Not only is this more intuitive, it also just seems easier to remember, as it uses things we should already know instead of being its own formula
auruxy
I had never thought about the determinant in this way, and it's definitely a much better and intuitive way of understanding what the determinant really means.
pw123
Does this mean we can expand this out like:
det(u,v,w) = (u x v).w = (transpose( (u_hat)(v) ))w?
xiaol3
This really is a very intuitive way to visualize determinants!
diegom
I've never actually been able to give a geometrical meaning to determinants so I really enjoyed this example. I looked into it and the youtube channel 3blue1brown has a few other incredible visualizations in his series on linear algebra... would highly recommend!
this is a much intuitive explanation than the diagonal rule
The geometic explanation really helps with the understanding of Triple Product
I think a little bit more about why this formula is the volume: from the geometric definition, (u cross v).w = |u cross v||w|cos, where |u cross v| is the area of the base, and |w|cos is the height of the parallelepiped since u cross v is orthogonal to the base.
Not only is this more intuitive, it also just seems easier to remember, as it uses things we should already know instead of being its own formula
I had never thought about the determinant in this way, and it's definitely a much better and intuitive way of understanding what the determinant really means.
Does this mean we can expand this out like: det(u,v,w) = (u x v).w = (transpose( (u_hat)(v) ))w?
This really is a very intuitive way to visualize determinants!
I've never actually been able to give a geometrical meaning to determinants so I really enjoyed this example. I looked into it and the youtube channel 3blue1brown has a few other incredible visualizations in his series on linear algebra... would highly recommend!