Quaternions: Extracting the Dot and Cross Products

The most important operations upon vectors include the dot and cross products and are indispensable for doing physics and vector calculus. The dot product gives a quick way to check whether vectors are orthogonal and the cross product calculates a new vector orthogonal to both its inputs. These vector operations were originally derived from the analysis of quaternions and eventually parted ways with quaternions to become our modern vector analysis. Much bickering ensued in the infancy of vector analysis, with the big names including Hamilton, Gibbs, and Heaviside, but we will focus on the math here.
In this video, I will show you how the dot and cross product arise from dissecting the formula for quaternion multiplication and allows us to write quaternion multiplication using dot and cross products. In addition, I will show the converse, how the dot and cross products can be defined using quaternion multiplication.

[@andrewhedman1271]

Awesome! I teach advanced precalculus and we're doing vector analysis now. I decided to motivate the cross-product this way today in class. I really like that it gives them historical context, something often missing from math discussions. We did the quaternion product and the kids were able to identify the parts of the resulting vector part, except for the cross product, of course. Without telling them what it was, we wrote it as a vector and showed using the dot product that it must be orthogonal to both vector parts of the original quaternions. We had a great discussion about what it means for a vector to be orthogonal to two other vectors, and the kids came to the correct conclusion that it's normal to the plane the two vectors are in. The hardest part for them to swallow was the non-commutativity of the products for i,j,k, so I gulped my coffee and we used my mug to have a quick discussion about rotations. Thanks for the idea!

[@hugojulianorellanarojas6762]

it's really impressive. Thank u so much.

[@TheDiverJim]

Hey Mathoma! I'm loving your video's on quaternions. They are hard for a guy like me to understand and you've done a great job explaining it. Would you PLEASE do a video on finding the angle between quaternions and the SLERP algorithm? I mentor a high school robotics team and someone who could explain this as well as you do, would be huge. Thanks for the great content.

[@MaxKawula]

well explained, very help

[@styfen90]

Thank you for this great video

[@ominollo]

Great explanation 👌

[@roberth1687]

Nice to see that link between quaternions and scalar/cross product. But it's not the only derivation of the scalar product. You could derive it through vector calculation as well (see: orthogonal projection).

[@sdmartens22]

From your geometric algebra videos, the quaternion multiplication restricted to scalars zero corresponds to the geometric product with the dot product negated and the cross product in place of the wedge product. Thanks for the great content, i was wondering what programs and audio hardware do you use to make your videos?

[@Math_oma]

+Shannon Martens Indeed, this is a good observation. The quaternions, as I'll get to in my geometric algebra series, are merely a subalgebra of G(3), namely the even subalgebra (scalars and bivectors only). Quaternions, complex numbers, split-complex numbers are all groping toward the same mathematical concept. I use an external microphone for my recent videos (compare to my old set theory videos), Open Broadcast Software to capture the screen and record audio, and SmoothDraw 4 with a Wacom tablet to write.

[@Math_oma]

And I use VideoPad to edit the video, but the editing is very rudimentary.

[@EngineerNick]

Thankyou I found this very helpful :)

[@harishgurusamy1044]

Nice vid!! However, I wondered how u extract the dot and cross products of octonions and split-octonions. I think it would be the same procedure with the quaternions but.....

[@TheNewPhysics]

What is the quaternion representation of a rotation in 4D (e.g. spacetime rotation)?

[@paulwolf3302]

Hey thanks, this whole subject is a little mind boggling but I understood this video beginning to end. When I start with Hamilton, he's applying it to electromagnetics, so the first thing ijk = -1 lost me right away. But as a general type of combined vector/scalar, not necessarily involving imaginary numbers, it's not hard to understand.

[@rupabasu4261]

Can anyone explain how the i,j,k componemts were taken in common to simplify the vector part

[@BlendingEdge]

Is there such thing as “Cross Product” of two quaternions ? Or is there only “Multiplication" of two quaternions which is defined as described in this video?

[@Math_oma]

+Blending Edge Cross product only makes sense in R^3 and quaternions are not R^3.

[@BlendingEdge]

@@Math_oma From what I gather the “Cross Product” makes sense only in 3 and 7 dimensions actually. (I didn’t expect that quaternions have a CP, btw, but only wanted to confirm that’s the case as I just started learning about quaternions).

[@janApen]

I'm in grade 12 and i'm loveing this.

[@1conscience0dimension]

and all this already existed by nature

[@jorgejimenez4325]

This is cool, but i don't see how it shows anything. How is this i-j-k business connected to the linear algebra we have now? How was this derivation of the cross product with these weird imaginary definitions translated into linear algebra, and how does this all connect back to perpendicular vectors and the area of parallelograms?