Cross Product

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Great question. I think the primary question is to ask, “When we have a force A, acting on a radius B, what is the resulting torque?”

We get |A| * |B| * sin(theta). [In my head, I see sine as a percentage – it shows what percent of the force will be applied, and ranges from 0% to 100%).

Ok. Now, how about the direction? Well, strictly speaking, we don’t need to create a direction vector, we could just leave the result as a scalar. But, if we want to pick a direction, what should we choose?

We could choose either A or B to be the primary direction, but why favor one over the other? (Is the force always the first term? What if we want B x A instead? How much should change in the result?)

A more generic description may be to have the axis of rotation be the result, and we can determine that we are spinning about this axis clockwise or counter-clockwise. This ends up being a mathematical convenience, but it seems the axis is a more neutral choice to describe the interaction.

We have cosine as a similarity measure
We have sine as a dis-similarity measure

What things in nature get better the more perpendicular they are? Twists, turns, spanning area, etc. General idea is “the more they move in their own direction, without overlap, the better we do.”

Two people working without getting in each other’s way. Find some scenario where being “across” is helpful (vs. being parallel).

We can define the cross-product as being the perpendicular vector, sure, but want to see why that calculation results in a perpendicular vector.

  • Intuition for why it’s not commutative [besides the calculation saying so… but it’s inverse, which is fine. We have a first, then second ordering. This goes positive or negative but the magnitude is the same.]

  • Intuition for why the cross product only works in 3d (and 7d)?

  • Ask: what setup are we trying to represent with a cross product?

To read:

Idea: derive the cross product from intersecting planes.

Do the cross product thing and write it out…

  • xyz (x cross y = z)… Going forward is positive, going backward is negative
  • Make the diagram and you can find the cross pretty easily.

Here’s the key intuition: our vectors are a combination of components. If we make something perpendicular to the components, it will be perpendicular to the WHOLE. Neat.

  1. Figure out easy ways to compute the perpendicular vector (each pair of components is voting on what is needed to make a vector perpendicular… or put another way, the perpendicular amount can be split to “cancel” each pair of components

  2. See how it’s satisfying two properties at once: one, the area of the parallelogram, and two, automatically becomes perpendicular. Neat. [Group theory on why this only works in R3 and R7?]

Core ideas:

  • How many ways can two vectors be combined?
  • Are there advantages to one way or another?

How can we combine vectors: what happens when they are multiplied?

  • Do they help or hurt each other when they are aligned? Fundamental question.
  • When spanning area, we want them perpendicular. When pushing together, we want them aligned.
  • Point of cross product (and dot product) is to label all the types of interaction two vectors can have.

Don’t see a mess of symbols… SEE what’s happening. We are getting a normal to each plane defined by the various vectors. We are moving that normal in a way that it’s always perpendicular to the various influences. Or rather, we imagine the two vectors being in their own plane (on the ground) and we are rotating into it. Hrm.

Good tutorial, also discusses quaternions:

Key is just seeing the possible interactions and realizing we want to account for them all.

  • Interactions between same components: dot product
  • Interactions between different components: cross product

$$\text{All interactions between } (a, b, c) \text{ and } (x, y, z) = \frac{1}{2}(\vec{a} \cdot \vec{b}) (\vec{a} + \vec{b}) + \vec{a} \times \vec{b} $$

Oh… the average “vector” times the dot product. Got it. Or, they each get one half the 'interaction effect".

Cross product interpretations:

  • Area swept out (or volume swept out) by following these vectors. It’s oriented though, so sweeping in a different direction matters.

Some scenarios work by having vectors in the same direction (projection). Some scenarios work by having vectors in perpendicular directions (computing volume of a prism). The cross product IS NOT the volume… it’s a computation that can calculate volume. Multiplication is NOT the “area”. It’s just a calculation that makes that easy.

Asking for the meaning of a cross product is like asking the meaning of a square root. We can visualize some scenarios where it’s useful, but it’s a computation. Use it or don’t. In some scenarios, that computation is how things behave: they work better when things are perpendicular.

  • Track all possible interactions
  • Dot product: interaction between similar elements
  • Cross product: interaction between different elements

Some scenarios, we want to track the interaction between similar elements (two vectors, how closely aligned are they? how similar these vectors?). Other scenarios, we need to track interaction between different elements: sweeping out area/volume. Putting force on a bar (torque). Etc. Have examples of this.

Q: Why is cross product perpendicular? Well, we could just leave it as an amount. But, if we put a perpendicular vector (why?) we can represent the plane of the area. But need a PROOF it’s perpendicular… that the dot product is zero.

Ah!!! Maybe that’s why it’s perpendicular… it has ALL the components which are not aligned. It must be perpendicular because all the “components in common” are not used!

ONLY the components which are not in common are used. Got it!!!

Proof orthogonal to factors:

Cross product in 2 dimensions (for complex numbers):

Get intuition for lagrange’s formula:’s_Formula

Intuition: We want to combine two vectors, and track:

  • their similarities: dot product
  • their differences: cross product

Similarity: some number which is a % that A is similar to B (use either, scale it up)
Differences: get a new vector which is DIFFERENT from the previous two. The differences are represented by a different vector, not a similar one.

$$|a||b|\sin(\theta)$$ just as a way to measure how different they are [compared to the MAX difference] and then, want this pointing in a direction that is different as well.

(Shouldn’t differences be represented in a different direction from the inputs?)

“Have all the interactions. The dot product is the interactions between the same elements, the cross product is interactions between different elements.”

Dot product: “Let the vectors interact, and measure their similarities”.

Cross product: “Let the vectors interact, and measure their differences.”

All interactions = similarities + differences = Dot product + Cross product

“Why do we need the cross product” -> “Why do we ever keep track of differences?”

You’ll think of a reason!!! Area/Volume is formed by motion in DIFFERENT dimensions. So track these differences and get the total volume formed.