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 be continued.


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

[2] W. Kaplan. Advanced Calculus (5th ed.). Addison-Wesley. 2002

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