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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.

*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.

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

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

Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! ЁЯЩВ https://www.patreon.com/patrickjmt !! Finding the Directional Derivative – In this video, I give the formula and do an example of finding the directional derivative that corresponds to a given angle. For more free math videos, visit http://PatrickJMT.com

Amazing, Life Changing Tutor
Before I met Jonathan, I was struggling through most of my STEM classes because I was simply not studying properly. He taught me all kinds of new study habits that would help me to save time, raise my grades, and lower my stress. Jonathan was specifically tutoring me in vector calculus and is an amazing tutor on the subject. You will walk away from a lesson with a game plan knowing exactly what you need to do before your next session with him or before your next exam in order to do well in the course. I highly recommend him as a tutor.

Amazing, Life Changing Tutor
Before I met Jonathan, I was struggling through most of my STEM classes because I was simply not studying properly. He taught me all kinds of new study habits that would help me to save time, raise my grades, and lower my stress. Jonathan was specifically tutoring me in vector calculus and is an amazing tutor on the subject. You will walk away from a lesson with a game plan knowing exactly what you need to do before your next session with him or before your next exam in order to do well in the course. I highly recommend him as a tutor.

From the authors that brought you the most successful calculus text published in the last two decades, Pearson is excited to present to you this interactive Precalculus eText with MyMathLab by Eric Schulz, William Briggs, Lyle Cochran. To learn more visit: www.pearsonhighered.com/schulzprecalcseries