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Given two vectors: $\mathbf{v}_1 = (2, 2)$ and $\mathbf{v}_2 = (3, 3)$:

Since the dot product is one method for measuring the similarity between vectors, and given that $\mathbf{v}_1 \cdot \mathbf{v}_1 = 8$ and $\mathbf{v}_1 \cdot \mathbf{v}_2 = 12$;

How can an identical vector have a lower dot product value than two different vectors?

Sammy Black
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Edmar Miyake
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    What is your question? Which vectors are identical? Both vectors point in the same direction but have different magnitudes. (Picture an arrow from the origin to the point $(2, 2)$ in the plane for $\mathbf{v}_1$ and a similar arrow but longer, reaching to $(3, 3)$ for $\mathbf{v}_2$.) – Sammy Black Apr 04 '24 at 00:52
  • Dot product is also proportional to vector length; cf. this question – J. W. Tanner Apr 04 '24 at 00:55

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The dot product of two vectors is the product of their lengths and the cosine of the angle between them. That's the only sense in which it "measures their similarity". Your intuition based on the magnitude of the dot product is wrong.

Ethan Bolker
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