An orthogonal transformation is a linear transformation 
on a finite-dimensional real vector space that preserves a positive
definite inner product. In particular, an orthogonal
transformation preserves lengths of vectors
and angles between vectors,
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(1)
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In three dimensions, an orthogonal transformation is either a rigid rotation or an improper
rotation (a rotation followed by a flip). If 
is the flip and 
the rotation, then flipping followed
by rotating has the form 
.
Thus it can also be realized by a suitable rotation
followed by the same flip, though in general this rotation
is not 
. Orthogonal transformations may be represented
using orthogonal matrices; their set
forms the orthogonal group.
Any linear transformation in three dimensions
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(2)
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(3)
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(4)
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satisfying the orthogonality condition
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(5)
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where Einstein summation has been used and 
is the Kronecker delta, is an orthogonal transformation.
If 
is an orthogonal transformation, then 
.
