Direction Cosines
Direction cosines of a vector refer to the cosines of the angles between the vector and the coordinate axes. For a three-dimensional vector , the direction cosines along the , and axes are given by:
\( \cos \alpha=\frac{v_1}{|\mathbf{v}|}, \cos \beta=\frac{v_2}{|\mathbf{v}|}, \cos \gamma=\frac{v_3}{|\mathbf{v}|}\)
Where:
- \( \alpha\) is the angle between the vector and the x -axis.
- \( \beta\) is the angle between the vector and the y-axis.
- \( \gamma\) is the angle between the vector and the z-axis.
- \( \gamma\) represents the magnitude (length) of the vector, calculated as \(\sqrt{v_1^2+v_2^2+v_3^2} . \).
These direction cosines help describe the orientation of a vector in three-dimensional mathematical and physical applications.
Direction Cosine with x-axis
Direction Cosine with y-axis
Direction Cosine with z-axis
Direction Cosine with all three axes