The mars polar lander spacecraft was launched on january 3, 1999. on december 3, 1999, the day that mars polar lander touched down on the martian surface, the positions of the earth and mars were given by these coordinates:
x y z
earth 0.3182 au 0.9329 au 0. au
mars 1.3087 au -0.4423 au -0.0414 au
in these coordinates, the sun is at the origin and the plane of the earth's orbit is the xy-plane. the earth passes through the +x -axis once a year on the autumnal equinox, the first day of autumn in the northern hemisphere (on or about september 22). one au, or astronomical unit, is equal to 1.496*10^8 km , the average distance from the earth to the sun.
1-find the distance from the sun to the earth in au on december 3, 1999.
2-find the distance from the sun to mars in au on december 3, 1999.
3-find the distance from the earth to mars in au on december 3, 1999.
4-as seen from the earth, what was the angle between the direction to the sun and the direction to mars on december 3, 1999?
5-explain whether mars was visible from your location at midnight on december 3, 1999. (when it is midnight at your location, the sun is on the opposite side of the earth from you.)