2) Consider the previously defined coordinate systems Fits and F Denote latitude by the symbol A and longitude by the symbol (1)/(2) Measure latitude positive north and negative south of the equator, -900leqslant lambda leqslant +90^circ Measure langitude positive east and negative west of zero longitude -180^circ lt mu leqslant +180^circ (a) Given the latitude A. and longitude A int _(0)^pi find a transformation matrix from F to F_(2)(T_(min )) that is a function only of A. and A. (b) Using your answer to part 2a transform the Earth's rotation vector in F_(C) to its representation in Earth-fixed coordinates at Blacksburg, Vinginla USA (F_(c)=F_(cm)) .That is, transform funday to omega _(m)) Use n_(man)=3712.442m mu _(max)=80-24.446m

(a) To find the transformation matrix from the coordinate system F to $F_2(T_{m\in})$, we need to consider the given latitude and longitude coordinates.<br /><br />The latitude is denoted by the symbol A and ranges from -90° to +90°, with positive values indicating north of the equator and negative values indicating south of the equator. The longitude is denoted by the symbol $\frac{1}{2}$ and ranges from -180° to +180°, with positive values indicating east of zero longitude and negative values indicating west of zero longitude.<br /><br />The transformation matrix from F to $F_2(T_{m\in})$ can be expressed as:<br /><br />$T_{m\in} = \begin{bmatrix} \cos(A) & \sin(A) & 0 \\ -\sin(A) & \cos(A) & 0 \\ 0 & 0 & 1 \end{bmatrix}$<br /><br />This matrix represents a rotation about the z-axis by an angle A (latitude) and a rotation about the y-axis by an angle $\frac{1}{2}$ (longitude).<br /><br />(b) To transform the Earth's rotation vector in $F_C$ to its representation in Earth-fixed coordinates at Blacksburg, Virginia, USA, we need to use the given values for the north-south and east-west components of the rotation vector.<br /><br />The north-south component is given by $n_{man} = 3712.442m$ and the east-west component is given by $\mu_{max} = 80-24.446m$.<br /><br />Using the transformation matrix from part (a), we can multiply the Earth's rotation vector in $F_C$ by the transformation matrix to obtain the representation in Earth-fixed coordinates:<br /><br />$\begin{bmatrix} \cos(A) & \sin(A) & 0 \\ -\sin(A) & \cos(A) & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} n_{man} \\ \mu_{max} \\ 0 \end{bmatrix} = \begin{bmatrix} n_{man}\cos(A) + \mu_{max}\sin(A) \\ n_{man}\sin(A) - \mu_{max}\cos(A) \\ 0 \end{bmatrix}$<br /><br />Therefore, the representation of the Earth's rotation vector in Earth-fixed coordinates at Blacksburg, Virginia, USA is:<br /><br />$\begin{bmatrix} n_{man}\cos(A) + \mu_{max}\sin(A) \\ n_{man}\sin(A) - \mu_{max}\cos(A) \\ 0 \end{bmatrix}$