# Questions tagged [vectors]

Use this tag for questions and problems involving vectors, e.g., in an Euclidean plane or space. More abstract questions, might better be tagged vector-spaces, linear-algebra, etc.

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### How to find the "average" direction of a set of vectors?

I have a series of vector directions and I need to find the "average" direction. I am not looking for the overall direction which would be the sum of the directions and this can't be used in cases ...
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### Finding the angle between vectors

What is the difference between $\cos(\theta) = \frac{(v \cdot u) }{( \|v\| \|u\| )}$ and $\cos(\theta) = \frac{|(v\cdot u)|}{( \|v\| \|u\| )}$ I noticed that my textbook uses simply the value of the ...
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### In what spaces are outer product of x with itself positive semidefinite?

In the case of vectors in $\mathbb R^n$, it is quite simple to see that for any vector $x$, $$v^Txx^Tv = (v^Tx)^2 \geq 0$$ so clearly the form $xx^T$ must form a positive semidefinite matrix. But ...
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### Does the functional square root of the cosine admit a vector-based interpretation?

In linear algebra, the cosine of the angle between two vectors $a$ and $b$ is defined as $$\cos(a,b) = \frac{\langle a, b \rangle}{||a||\cdot||b||} .$$ The functional square root of the cosine has at ...
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### Is there a version of tuples without duplicates?

A set is a list of elements that is unordered and does not permit duplicates, so: $$\{1,2\} = \{2,1\}$$ $$\{1,2\} = \{1,2,2\}$$ A bag or multiset is unordered, but it allows duplicates, so: [1,2]=[2,...
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