Vector Calculus: Understanding the Cross Product

This is a companion discussion topic for the original entry at

Kalid has done it again! Imagine if we are taught like this in secondary schools, our university engineering education would have been so fantastic

Thanks Theophilus, glad you enjoyed it! I really wish someone showed me how the dot and cross product combine to build out the full picture.

Yeah, very great explanatory series. Very helpful and intuitive. Unfortunately this is often not a norm even at universities. Thanks much :wink:

I create two triangles with numbers at their edges, in order to represent two 3D vectors. I use them as base and top of a triangular cylinder. I draw two diagonal lines on each sides of the cylinder. Each of the 6 diagonal lines gets a number. Since I made my triangular representation of vector follow a counter-clockwise pattern, diagonal lines that go down, or approaches the base, as I go counter-clockwise gets (+) sign, and the diagonal lines that go up as I go counter-clockwise gets (-) sign. Each set of 3 pairs of diagonal lines crossing each other generates a number, giving us 3 pieces of information.

A perplexing point to me is that we choose to store the number we get by observing interaction of (y&z) and (z&y) into x! My childish guess is that x is neutral to y and z, so both y and z are happy to have their votes stored in x’s chest.

Actually this is the completely wrong way to teach this. In actual fact it’s actually properly taught by using Geometric Algebra. I.e clifford algebra.
Using the “proper way” leads to far more insight and intuition into how this is used and what it really means in actual use.
It’s a shame that it’s never taught properly.

@Mike: Glad you enjoyed it!

@Jae: I might need to see a diagram, but yep – the idea is that x is perpendicular (not favoring) either y or z.

@Haniff: The idea is to show how the traditionally taught dot/cross product fit into a larger picture, but not to start with geometric algebra on day 1 (“I thought we were in a vector calculus class?”).

Kalid ,well done again, can we see result of complex multiplication in terms of total contribriution of components,I mean complex multiplication is multiplication of x and y coordinates, where similar coordinates multiplication indicates the total push (dot product) and different coordinates multiplication indicate magnitude (area,volume;cross product) and then we subtract the push which is useless( i square),plz comment ,thanks once again for insight

Ever since my kids came home from school with the “lattice method” of multiplication and it took me a score of tries to “get it”, the true insight of using the distributive property to break numbers into smaller (or different) components has fascinated me. Kalid does it again with this nice graphic. Can’t wait for the day when he and Vihart are in the same real (or virtual) space.

Thanks Mark, glad the matrix/lattice clicked. You might like this one too:

I’d love to collaborate with Vi one day.

Great article but I’m a little confused on why the dot product is a number whereas the cross product is a vector. Could you elaborate further please? Also, off the topic, do you think you’ll ever do an abstract algebra section?


Hi Jackson, great question. The dot product measures similarity (how similar is vector A to vector B?), and the result is a percentage. The assumption is you already have vector B, so a percentage is enough. If you like, you can do “(A dot B) times B” to get an actual vector.

The cross product asks “How different are A and B?”. Although the difference can be quantified as a percentage, there is ambiguity because two vectors can be entirely different from a given vector. (I.e., North and South are entirely different from East). Specifying the cross product as a vector means we can distinguish “The difference between North and East” from “The difference between South and East”.

Abstract algebra would a fun topic down the line, thanks for the suggestion.


mr. kalid simply you are a geneous of math. but on the contrary i am really a foolish one.

Please answer my question

This is a fantastic explanation and answer. I finally have a fundamental understanding and knowledge of the cross product after all these years. This is well-written and it all makes sense to me now. I will be coming back to this site again for more articles like these after what I’ve learnt today. Thank you.


w=fa why scalar quantities

Thanks Jake, glad you enjoyed it. Hope you enjoy the rest of the site.

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