Coordinate System: Rotating 2D & 3D Axes & Impact on Directed Cosines

[paragchitnis]

Co-ordinate axes can be drawn on plane or in space. In case of plane geometry (2D-geometry) the two axes, X-axis and Y-axis, can be rotated through some angle. For this transformation of axes we get the relation between the co-ordinates of a point in old and new system. Is this transformation is possible in 3D geometry (Rotation of X,Y,Z-axes)? If yes, then will it affect the numerical value of Directed Cosines of a fixed line?

 

[jambaugh]

Yes, of course.
One way to think of the direction cosines is to consider the dot product of the (unit) direction vector in the direction of the line with the unit coordinate vectors which gives the coordinate values:

$$\mathbf{\hat{v}}\cdot\mathbf{\hat{\imath}} = \cos(\theta_x) = v_x$$
$$\mathbf{\hat{v}}\cdot \mathbf{\hat{\jmath}} = \cos(\theta_y) = v_y$$
$$\mathbf{\hat{v}}\cdot \mathbf{\hat{k}} = \cos(\theta_z) = v_z$$

Rotations of the axes affects the 3 components via multiplication by a rotation matrix and so the same matrix will transform the directional cosines.

 

[paragchitnis]

Thank you very much