A unit vector is a vector with a magnitude of 1. The unit vector $ \mathbf{\hat{v}} $ is defined by

$ \mathbf{\hat{v}}=\frac{\mathbf{v}}{\|\mathbf{v}\|} $

where $ \mathbf{v} $ is a non-zero vector.

$ \mathbf{\hat{i}},\mathbf{\hat{j}},\mathbf{\hat{k}} $ are commonly used as the unit vectors forming a basis in the $ x,y,z $ directions respectively. They are often used as notation for vectors or in vector functions:

$ \vec{F}(t)=x(t)\mathbf{\hat{i}}+y(t)\mathbf{\hat{j}}+z(t)\mathbf{\hat{k}} $

Another common set of unit vectors forming a basis is in the Frenet–Serret formulas, where $ \mathbf{\hat{T}},\mathbf{\hat{N}},\mathbf{\hat{B}} $ represent the tangent, normal, and binormal vectors respectively.

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