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Why can't you multiply vectors?

Freya Holmér · talk · Watch on YouTube

Pure mathematicsRecreational & visual mathGame programming & engines

What it is

Freya Holmer's Dutch Game Day 2023 talk asking why vectors can't be multiplied, building from complex numbers to a single axiom and arriving at geometric algebra, quaternions and multivectors.

Why Hacker News recommends it

HN treats it as the best entry point to geometric algebra and an intuitive route to understanding quaternions.

  • 9 people recommended it
  • 5 different months
  • 5 strong recommendations
  • first 2023, last 2025

What HN commenters said

  1. “I highly recommend anyone who finds this stuff interesting but confusing (like I do) to check it out”

    Fascinating, entertaining talk deriving quaternions from one axiom about vector multiplication

    mkaic, 2023-10 · Strong recommendation · in “Exploring Quaternions (2018)”

  2. “I highly recommend a recent talk by Freya Holmer.”

    Fun exploration of vector multiplication culminating in an introduction to Clifford algebras

    Syntonicles, 2023-10 · Strong recommendation · in “Linear Algebra Done Right – 4th Edition”

  3. “One of my favorite math/graphics YouTube creators Freya Holmér did an excellent intro to Geometric Algebra”

    Excellent intro to geometric algebra from a favorite math/graphics creator

    zoogeny, 2024-02 · Strong recommendation · in “Look, ma, no matrices”

  4. “Freya did an excellent job reducing the number of axioms to 1”

    Reduces the axioms to one: a vector squared equals its length squared

    simpaticoder, 2024-02 · Strong recommendation · in “Look, ma, no matrices”

  5. “The absolute first video to watch should be the (ironically titled) Why can't you multiply vectors?”

    The first video to watch on geometric algebra

    gowld, 2024-08 · Strong recommendation · in “Poor Foundations in Geometric Algebra”

  6. “this conference talk[1] released yesterday regarding vector multiplication”

    Explains quaternions as an extension of 3D vector multiplication via bivectors

    conradludgate, 2023-10 · Recommendation · in “Exploring Quaternions (2018)”

  7. “a good place to jump in might be Freya Holmér's Why can't you multiply vectors?”

    Good entry point into geometric algebra

    infogulch, 2024-02 · Recommendation · in “Rapier is a set of 2D and 3D physics engines written in Rust”

  8. “it is about multiplying vectors and what you get from that, and it includes quaternions.”

    Talk on multiplying vectors that includes quaternions as a case

    GuB-42, 2024-04 · Recommendation · in “Quaternion Knowledge Graph Embeddings (2019)”

Excerpts from Hacker News comments, linked to the originals. Summaries and labels were written by a language model from those comments.

Topics: vector multiplication, quaternions, clifford algebra, complex numbers, game math

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