100. Navigating in the Solar System. 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:
$\begin{array}{llll}\text { Earth } & 0.3182 \mathrm{AU} & 0.9329 \mathrm{AU} & 0.0000 \mathrm{AU} \\ \text { Mars } & 13087 \mathrm{AU} & -0.4423 \mathrm{AU} & -0.0414 \mathrm{AU}\end{array}$
In these coondinates, the sun is at the origin and the plane of the earth's orbit is the $x y-$ plane. The earth passes through the $+x$-xis 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 eqnal to $1.496 \times 10^{8} \mathrm{~km}$, the average distance from the earth to the sun. (a) In a diagram, show the positions of the sun, the earth, and Mars on December 3,1999 .(b) Find the following distances in AU on December 3,1999 : (i) from the sun to the earth; (ii) from the sun to Mars; (iii) from the earth to Mars. ( $c$ ) As seen from the earth, what was the angle between the direction to the sun and the direction to Mars on December $3,1999 ?$ (d) 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.)