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The term unit vector pops up sometimes and I'm having quite a time knowing what it is to be utilized for.

A unit vector represents direction as it's length is 1 and leaves scalar multiplication alone(or rather doesn't interfere with scalar multiplication).

A quote I found:

Using unit vectors instead of vectors of varying lengths is generally the preferred way to do any vector math, since it will only take into account the vector's direction, not its magnitude.

For example, pattern recognition utilizes linear algebra. So does the Google search engine to represent page rank. What would unit vectors be used for. I normally require a reason to fully grip the term and be able to visualize the possibilities and in what types of domains, scopes it can be utilized in.

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Given: a force of magnitude $\left \| \mathbf{F} \right \| = 1 [N]$ in the direction of $v = (1,2,1)^T$.

Question: what is $\mathbf{F}$ ?

Answer:

\begin{align*} \mathbf{F} & = \left \| F \right \| \frac{1}{\left \| v \right \|} v\\ & = \frac{1}{\sqrt{6}} \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} [N] \end{align*}

([N] = [Newton])

As you can see, not normalizing would lead to a force vector of magnitude $\sqrt{6} [N]$ (which is too much).

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