Cross products in the light of linear transformations | Chapter 11, Essence of linear algebra

Why the formula for cross products matches the geometric intuition. Help fund future projects:   / 3blue1brown   An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ For anyone who wants to understand the cross-product more deeply, this video shows how it relates to a certain linear transformation via duality. This perspective gives a very elegant explanation of why the traditional computation of a dot product corresponds to its geometric interpretation. Minor error at 1:44, the third line of the matrix should read "v1 * w2 - w1 * v2" *Note, in all the computations here, I list the coordinates of the vectors as columns of a matrix, but many textbooks put them in the rows of a matrix instead. It makes no difference for the result since the determinant is unchanged after a transpose, but given how I've framed most of this series I think it is more intuitive to go with a column-centric approach. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter:   / 3blue1brown   Patreon:   / 3blue1brown   Facebook:   / 3blue1brown   Reddit:   / 3blue1brown