Before getting back to the lectures from 3Blue1Brown, try this part review, part preview:
Now let's get back into the meaning with a little more detail.
A9. The dot (or scalar) product
The dot product is a way to estimate how much two vectors interact in a common dimension. If the vectors are orthogonal to each other, they don't interact in a common dimension so their dot product is zero. This is like asking how much north-south movement is involved in an east-west heading: none. But if two vectors are perfectly parallel then this is equivalent to the two vectors lying on the number line and we can use our standard (scalar) multiplication rules. In between, we use a little trigonometry to determine their (dot) product.
Still lacking an intuition? This excellent summary from Better Explained (slogan: "Learn Right, Not Rote") should do the trick.
A10. The cross (or vector) product
Both the dot and cross products affect dimensionality. With the dot product, we find how much two vectors interact in one dimension. The cross product of two vectors is perpendicular to them both, telling us how much rotation arises in a third dimension.
A useful real world example use of the cross product is to compute the torque vector. Torque is the rotating force generated by pulling or pushing on a lever, such as a wrench or a bicycle crank. The lever moves in one plane but produces a rotation orthogonal to that plane.
Torque is also fundamental to the origins of the MRI signal. We will encounter it later in the physics section. Can you take a guess how torque might be relevant to the MRI signal? Hint: it has to do with the interaction of a nuclear magnet (the protons in H atoms) with an applied magnetic field.
This article from Cuemath covers the rules for computing dot and cross products. And here are a couple of useful visualizations:
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