Dot product and Duality

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Dot product

In similar directions v*w > 0

In perpendicular direction v*w = 0

In Opposing directions v*w < 0

Dot product is translate one of the vector to the transformation.

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Duality

Making a symmetric line which seperate basis vector and input vector equally.

When you try find a linear transformation to map vectors in 2-D space into number line, you will be able to match it to some vector, which is called dual vector of the transformation.

Reference
https://www.3blue1brown.com/topics/linear-algebra