## Friday, September 23, 2011

### Math - Vectors - Unit vector and vector magnitude/length/norm

The unit vector of a vector is in the direction of the vector with a magnitude of one.  For example, in the three-dimensional Cartesian coordinate system the vectors $$\left( \hat{i}, \hat{j}, \hat{k} \right)$$ or $$\left( \mathbf{i}, \mathbf{j}, \mathbf{k} \right)$$ represent the unit vectors along the three axes $$\left(x, y, z \right)$$.

In general, the unit vector of a vector can be written as

$\mathbf{e}_A = \dfrac{ \mathbf{A}}{| \mathbf{A}|}$

which is the vector of $$A$$ divided by the magnitude of $$A$$.  The magnitude (also called the length or norm) in the three-dimensional case $$\vec{A} = A_1 \mathbf{i} + A_2 \mathbf{j} + A_3 \mathbf{k}$$ is expressed as

$| \mathbf{A}| = \sqrt{\mathbf{A} \cdot \mathbf{A}} = \sqrt{A_1^2 + A_2^2 + A_3^2}$

In an orthogonal right-handed system, the unit vectors in three dimensions can be expressed as $$\mathbf{e}_1$$, $$\mathbf{e}_2$$, and $$\mathbf{e}_3$$ where the components of a vector $$\mathbf{A}$$ are $$A_1$$, $$A_2$$, and $$A_3$$ and correspond to the unit vectors respectively.  That is the vector $$\mathbf{A}$$ can be broken down into individual components which relate to whatever coordinate system and unit vectors are chosen to be used [1, 2].

In progress...to be continued.

References:

[1] K. Karamcheti. Principles of Ideal-Fluid Aerodynamics. John Wiley & Sons, Inc., New York, NY. 1966