Versor (physics)

Versors i, j, k of the Cartesian axes x, y, z for a three-dimensional Euclidean space. Every vector a in that space is a linear combination of these versors.

In geometry and physics, the versor of an axis or of a vector is a unit vector indicating its direction.

The versor of a Cartesian axis is also known as a standard basis vector. The versor of a vector is also known as a normalized vector.

Versors of a Cartesian coordinate system

The versors of the axes of a Cartesian coordinate system are the unit vectors codirectional with the axes of that system. Every Euclidean vector a in a n-dimensional Euclidean space ( $R n$ ) can be represented as a linear combination of the n versors of the corresponding Cartesian coordinate system. For instance, in a three-dimensional space ( $R$ ³), there are three versors:

{\mathbf {i}}=(1,0,0),

{\mathbf {j}}=(0,1,0),

{\mathbf {k}}=(0,0,1).

They indicate the direction of the Cartesian axes x, y, and z, respectively. In terms of these, any vector a can be represented as

{\mathbf {a}}={\mathbf {a}}_{x}+{\mathbf {a}}_{y}+{\mathbf {a}}_{z}=a_{x}{\mathbf {i}}+a_{y}{\mathbf {j}}+a_{z}{\mathbf {k}},

where a_x, a_y, a_z are called the vector components (or vector projections) of a on the Cartesian axes x, y, and z (see figure), while a_x, a_y, a_z are the respective scalar components (or scalar projections).

In linear algebra, the set formed by these n versors is typically referred to as the standard basis of the corresponding Euclidean space, and each of them is commonly called a standard basis vector.

Notation

A hat above the symbol of a versor is sometimes used to emphasize its status as a unit vector (e.g., ${\hat {{\mathbf {\imath }}}}$ ).

In most contexts it can be assumed that i, j, and k, (or ${\vec {\imath }},$ ${\vec {\jmath }},$ and ${\vec {k}}$ ) are versors of a 3-D Cartesian coordinate system. The notations $({\hat {{\mathbf {x}}}},{\hat {{\mathbf {y}}}},{\hat {{\mathbf {z}}}})$ , $({\hat {{\mathbf {x}}}}_{1},{\hat {{\mathbf {x}}}}_{2},{\hat {{\mathbf {x}}}}_{3})$ , $({\hat {{\mathbf {e}}}}_{x},{\hat {{\mathbf {e}}}}_{y},{\hat {{\mathbf {e}}}}_{z})$ , or $({\hat {{\mathbf {e}}}}_{1},{\hat {{\mathbf {e}}}}_{2},{\hat {{\mathbf {e}}}}_{3})$ , with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity. This is recommended, for instance, when index symbols such as i, j, k are used to identify an element of a set of variables.

Versor of a non-zero vector

The versor (or normalized vector) ${\hat {{\mathbf {u}}}}$ of a non-zero vector $\mathbf {u}$ is the unit vector codirectional with $\mathbf {u}$ :

{\hat {{\mathbf {u}}}}={\frac {{\mathbf {u}}}{\|{\mathbf {u}}\|}}.

where $\|{\mathbf {u}}\|$ is the norm (or length) of $\mathbf {u}$ . Notice that a versor lost the units of the original vector. For instance, if we have the vector ${\vec {u}}=(0,5,0)\,\mathrm {m}$ , then $|{\vec {u}}|={\sqrt {0^{2}+5^{2}+0^{2}}}\,\mathrm {m} =5\,\mathrm {m}$ and

${\hat {u}}={\frac {\vec {u}}{|{\vec {u}}|}}={\frac {(0,5,0)\,\mathrm {m} }{5\,\mathrm {m} }}=(0,1,0)$

You can notice that ${\hat {u}}$ is a dimensionless quantity.

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