Direction Cosine

In 3-dimension (3-D) space, a non-zero vector can be denoted by

\begin{align*} \mathbf{v} = v_x \mathbf{e}_x + v_y \mathbf{e}_y + v_z \mathbf{e}_z, \end{align*}

where

• \(\mathbf{e}_x = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\), \(\mathbf{e}_y = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\), and \(\mathbf{e}_z = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\) are the unit vectors of the 3 axes, respectively.
• \(v_x\), \(v_y\), and \(v_z\) are the components of \(\mathbf{v}\) projected to the 3 axes.

For vector \(\mathbf{v}\), its direction cosine vector can be expressed as

\begin{align*} \begin{bmatrix} \cos\alpha \\ \cos\beta \\ \cos\gamma \end{bmatrix} &= \begin{bmatrix} \dfrac{v_x}{\| \mathbf{v}\|_2} \\ \dfrac{v_y}{\| \mathbf{v}\|_2} \\ \dfrac{v_z}{\| \mathbf{v}\|_2} \end{bmatrix} \\ &= \dfrac{\mathbf{v}}{\| \mathbf{v}\|_2} \\ &\triangleq \mathbf{e}_v, \end{align*}

where

• \(\alpha\), \(\beta\), \(\gamma\) are the angles of vector \(\mathbf{v}\) relative to the 3 axes, respectively.
• \(\mathbf{e}_v\) is the unit vector of \(\mathbf{v}\), i.e., \(\| \mathbf{e}_v \|_2 = \sqrt{\cos^2\alpha + \cos^2\beta + \cos^2\gamma} = 1\).