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49 lines
1.1 KiB
Markdown
49 lines
1.1 KiB
Markdown
---
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cards-deck: default_obsidian
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date: [[2022-04-27]]
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tags: #linear_algebra
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---
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#math/linear_algebra
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# Dot Product
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multiply each number in a vector by the one in the corresponding index in the other. then add it all up.
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## Length of vector
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#card
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dot product with itself, square root.
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it's the pythagorean theorem.
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$$||v||=\sqrt{v\cdot v}$$
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^1651362768047
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## unit vector
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#card
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$$||cv|| = |c|\ ||v||$$
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vector divided by its length, you get a vector of length 1
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^1651363913154
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## Distance:
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#card
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$$dist(u,w)=||u-w||$$
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you are basically subtracting the vectors, and doing pythagorean to get the length of that resulting vector
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^1651363193147
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## **cauchy-schwartz**
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#card
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$$|u\cdot w|\leq ||u||\ ||w||$$
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basically, the dot product is always smaller than the two lengths multiplied
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^1651363193158
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## **triangle inequality**
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#card
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$$||u+w||=||u||+||w||$$
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the length of the added vectors is always smaller than the added lengths of the vectors.
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^1651363193169
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## **angles?**
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%%
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#card
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%%
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$$cos\theta = \frac{uv}{||u||\ ||v||}=\frac{u}{||u||}\cdot \frac{v}{||v||}$$
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^1651363426724
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